Introduction:
Free-form curves and surfaces where developed to describe curved 3-D objects without using polyhedral representations which are bulky and intractable. To get a precise curve with polygons might require thousands of faces, whereas a curved surface requires much less calculations.
The development of Free-form curves and surfaces for computer graphics begins late 60s with, P. de Casteljau (Citroen) and P. Bezier (Renault), engineers in the french auto industry. P. de Casteljau was earlier than Bezier, but was never published so Bezier get mosts of the credit. These men were pioneers in Computer Aided Geometric Design (CAGD) for the auto industry, which replaced the use of hand drawn french curve templates in design of auto bodies.
The major breakthroughts in in CAGD were the theory of Bezier surfaces and Coons patches which were later combined with B-spline methods.
Important Terms:
B-spline - A piecewise polynomial function. A spline curve is expressed in terms of B-splines.
It is easier to piece curves together using B-splines rather then Bezier curves because they use a set of blending functions that have local support only - the location of the curve depends on only a few control points. Contributors to B-spline development: Ahuja, Catmull and Rom, Clark, Coons, Cox, deBoor, Forest - 1970s.
Beta-splines -
Bezier Curves - polynomial curves expressed in terms of Berstein polynomials. Part of the UNISURF systemd developed by P. Bezier.
Based on a family of functions called Bernstein polynomials.
Polynomial Curves - includes Bezier Curves and NURBs
NURBs - Nonuniform rational B-spline curves
Computer Aided Geometric Design (CAGD) - pioneered by P. de Casteljau (Citroen) and P. Bezier (Renault), auto designers in the 1960s.
control points - a common way of controlling the shape of a curves in a predicable way
piecewise - a function describing only a small piece of a curve
rational polynomials - defined by the algebraic ratio of two polynomial functions. Developed by Rowin, Roberts, Coons, and Forrest in the 1960s.
Surface Patches - the locus of all points of a moving or defroming curve. A surface is often broken down into patches.First developed by Coons.
others - quadric surfaces, natural splines, Coons surfaces, etc.
Significant Contributors notes:
curves and surfaces SIGGRAPH Bibliography
name - location of research - contribution - years
P. de Casteljau (Citroen) and P. Bezier (Renault) - created the UNISURF system (1972) for auto design
1970 - Pierre Bezier from Renault develops Bezier free-form curve representation
A. Ball - Computer Aided Design 1970s, 80s
R. Barnhill - Computer Aided Geometric Design 1970s, 80sB. Barsky - University of Utah, 1980s
S. Bernsteien - Mathematician, Polynomials, early 1900s - The Bernstein Form of a Bezier Curve - nonrecursive formula
W. Boehm - Generating the Bézier Points of B-Spline Curves and Surfaces, Smooth rational curves, An affine representation of de Casteljau's and de Boor's rational algorithms, Aspects of Car Body Design at Daimler-Benz, Cubic B-Spline Curves and Surfaces in Computer-Aided Geometric Design
Computer Aided Geometric Design 1970s, 80s
E. Catmull - Computer Aided Geometric Design 1970s
1974 - Alex Schure opens CGL at NYIT, with Ed Catmull as Director
1974 - z-buffer developed by Ed Catmull (Univ of Utah)
1975 - Catmull curved surface rendering algorithm A Subdivision Algorithm for Computer Display of Curved Surfaces, (December 1974)
G. Chang - Computer Aided Geometric Design 1980s
E. Cohen - 1980s
S. Coons - parametric cubic curve and bicubic surface interpolating techniques, Surface Patches and B-Spline Curves, 1960s, 70s
1963 - Coons' patches
ACM SIGGRAPH Steven A. Coons Award
P. Davis - University of Utah - The convexity of Bernstein polynomials over triangles, 1970s
C. de Boor - B-spline development, Piecewise polynomial interpolation and approximation, Bicubic spline interpolation, 1960s - 80s
G. de Rham - 1940s, 50s
T. DeRose - 1980s
A. Forrest - The twisted cubic curve: a computer-aided geometric design approach, Interactive interpolation and approximation by Bezier polynomials, 1960s -1980s
G. Farin - Daimler-Benz - Algorithms for Rational Bézier Curves, Smooth curves and surfaces, 1980s
D. Ferguson - parametric cubic curve and bicubic surface interpolating techniques, Surface shape control using constrained optimization on the B-spline representation, Construction of curves and surfaces using numerical optimization techniques, 1960s - 1980s.
Faux and Pratt - advantages of higher order Bezier curves, 1980s
R. Goldman. Some Properties of Bézier Curves, 1980s
T. Goodman - 1980s
W. Gordon - 1960s, 70s
J. Gregory - 1970s, 80s
J. Hoschek - 1980s
G. Nielson - 1970s, 80s
Les A. Piegl - Filling n-sided regions with NURBS patches, 1990s
1973 - Principles of Interactive Computer Graphics (Newman and Sproull) first comprehensive graphics textbook is published
Search Term: curves and surfaces
#1:
Dinesh Manocha and John Canny. A New Approach for Surface Intersection, SMA '91: Proceedings of the First Symposium on Solid Modeling Foundations and CAD/CAM
Applications, pp. 209-220 (June 1991, held June 5-7, 1991 in Austin, Texas, USA. ). ACM. Edited by Jaroslaw Rossignac and Joshua Turner. ISBN 0-89791-427-9.
Abstract
Evaluating the intersection of two rational parametric surfaces is a recurring operation in solid modeling. However, surface intersection is not an easy problem and continues to
be an active topic of research. The main reason lies in the fact that any good surface intersection technique has to balance three conflicting goals of accuracy, robustness and
efficiency. In this paper, we formulate the problems of curve and surface intersections using algebraic sets in a higher dimensional space. Using results from Elimination theory,
we project the algebraic set to a lower dimensional space. The projected set can be expressed as a matrix determinant. The matrix itself, rather than its symbolic determinant, is
used as the representation for the algebraic set in the lower dimensional space. This is a much more compact and efficient representation. Given such a representation, we utilize
properties of straight line programs and results from linear algebra for performing geometric operations on the intersection curve. Most of the operations involve evaluating
numeric determinants and computing the rank, kernel and eigeiivalues of matrices. The accuracy of such operations can be improved by pivoting or other numerical techniques.
We use this representation for inversion operation, computing the intersection of curves and surfaces and tracing the intersection curve of two surfaces in lower dimension.
Suggested/Internal Citation Key
Manocha:1991:ANA
#2:
Dinesh Manocha. Solving polynomial systems for curve, surface and solid modeling, SMA '93: Proceedings of the Second Symposium on Solid Modeling and Applications,
pp. 169-178 (May 1993, held May 19-21, 1993 in Montreal, Quebec, Canada). ACM.
Abstract
Current geometric and solid modeling systems use semi-algebraic sets for defining the boundaries of solid objects, curves and surfaces, geometric constraints with mating
relationship in a mechanical assembly, physical contacts between objects, collision detection. It turns out that performing many of the geometric operations on the solid
boundaries or interacting with geometric constraints is reduced to finding common solutions of the polynomial equations. Current algorithms in the literature baaed on symbolic,
numeric and geometric methods suffer from robustness, accuracy and efficiency problems or are limited to a class of problems only, In this paper we present algorithms based
on multipolynomial resultants and matrix computations for solving polynomial systems arising in modeling applications. These algorithms are based on the linear algebra
formulation of resultants of equations and in many cases there is an elegant relationship between the matrix structures and the geometric formulation. The resulting algorithm
involves matrix computations and in the context of floating point computation their numerical accuracy is well understood. We also present techniques to make use of the
structure of the matrices to improve the performance of the resulting algorithm and highlight the performance of the algorithms on boundary computations.
Suggested/Internal Citation Key
Manocha:1993:SPS
#3:
S. S. Abhyankar and C. J. Bajaj. Automatic Parameterization of Rational Curves and Surfaces IV: Algebraic Space Curves, ACM Transactions on Graphics, 8 (4), pp.
325-334 (October 1989).
Copyright
Copyright © 1989 Association for Computing Machinery
Suggested/Internal Citation Key
Abhyankar:1989:APO
#4:
Barry Joe. Knot insertion for beta-spline curves and surfaces, ACM Transactions on Graphics, 9 (1), pp. 41-65 (January 1990). ISSN 0730-0301.
Copyright
Copyright © 1990 Association for Computing Machinery
Suggested/Internal Citation Key
Joe:1990:KIF
#5:
Ari Rappoport. Rendering Curves and Surfaces with Hybrid Subdivision and Forward Differencing, ACM Transactions on Graphics, 10(4), pp. 323-341 (October
1991). ISSN 0730-0301.
Keyword(s)
adaptive forward differencing, Bézier curves and surfaces, forward differencing, parametric curves and surfaces, subdivision methods
Copyright
Copyright © 1991 Association for Computing Machinery
Abstract
We present a Hybrid Rendering Algorithm (HRA) for rendering parametric curves and surfaces. The algorithm uses a series of Direct Rendering Criteria (DRC) for
determining whether the curve surface can be directly rendered by forward differencing with a constant step size. The DRCS test the geometric flatness of the curve/surface, its
parametric uniformity, and the ability to use only integer arithmetic in the forward differencing algorithm. If any of the DRCS is not fulfilled, the curve, surface is subdivided,
The location of the subdivision in parameter space is chosen to increase the chances that the new segments will satisfy the DRCS, For the integer arithmetic DRC we introduce a
general method for determining an alignment of tbe forward differences. We show that for cubic [quartic) curves whose control points lie in a 128K ? 128K space this alignment
enables up to 213(211) forward steps. The method is applicable to curves of any order.
Suggested/Internal Citation Key
Rappoport:1991:RCA
#6:
Fuhua Cheng. Estimating Subdivision Depths for Rational Curves and Surfaces, ACM Transactions on Graphics, 11(2), pp. 140-151 (April 1992). ISSN 0730-0301.
Copyright
Copyright © 1992 Association for Computing Machinery
Abstract
An algorithm to estimate subdivision depths for rational curves and surfaces is presented. The subdivision depth is not estimated for the given curve/surface directly. The
algorithm computes a subdivision depth for the polynomial curve/surface of which the given rational curve/surface is the image under the standard perspective projection. This
subdivision depth, however, guarantees the required flatness of the given curve/surface after the subdivision. This work has applications in surface rendering, surface/surface
intersection, and mesh generation.
Suggested/Internal Citation Key
Cheng:1992:ESD
#7:
Demetri Terzopoulos and Hong Qin. Dynamic NURBS with Geometric Constraints for Interactive Sculpting, ACM Transactions on Graphics, 13(2), pp. 103-136 (April
1994). ISSN 0730-0301.
Copyright
Copyright © 1994 Association for Computing Machinery
Abstract
This article develops a dynamic generalization of the nonuniform rational B-spline (NURBS) model. NURBS have become a defacto standard in commercial modeling systems
because of their power to represent free-form shapes as well as common analytic shapes. To date, however, they have been viewed as purely geometric primitives that require
the user to manually adjust multiple control points and associated weights in order to design shapes. Dynamic NURBS, or D-NURBS, are physics-based models that
incorporate mass distributions, internal deformation energies, and other physical quantities into the popular NURBS geometric substrate. Using D-NURBS, a modeler can
interactively sculpt curves and surfaces and design complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and
weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. D-NURBS move and deform in a physically
intuitive manner in response to the user's direct manipulations. Their dynamic behavior results from the numerical integration of a set of nonlinear differential equations that
automatically evolve the control points and weights in response to the applied forces and constraints. To derive these equations, we employ Lagrangian mechanics and a
finite-element-like discretization. Our approach supports the trimming of D-NURBS surfaces using D-NURBS curves. We demonstrate D-NURBS models and constraints in
applications including the rounding of solids, optimal surface fitting to unstructured data, surface design from cross sections, and free-form deformation. We also introduce a
new technique for 2D shape metamorphosis using constrained D-NURBS surfaces.
Suggested/Internal Citation Key
Terzopoulos:1994:DNW
#8:
Ari Rappoport and Yaacov Hel-Or and Michael Werman. Interactive Design of Smooth Objects with Probabilistic Point Constraints, ACM Transactions on Graphics, 13(2),
pp. 156-176 (April 1994). ISSN 0730-0301.
Copyright
Copyright © 1994 Association for Computing Machinery
Abstract
Point displacement constraints constitute an attractive technique for interactive design of smooth curves, surfaces, and volumes. The user defines an arbitrary number of "control
points" on the object and specifies their desired spatial location, while the system computes the object's degrees of freedom so that the constraints are satisfied. A
constraint-based interface gives a feeling of direct manipulation of the object. In this article we introduce soft constraints, constraints which do not have to be met exactly. The
softness of each constraint serves as a nonisotropic, local shape parameter enabling the user to explore the space of objects conforming to the constraints. Additionally, there is a
global shape parameter which determines the amount of similarity of the designed object to a rest shape, or equivalently, the rigidity of the rest shape. We present an algorithm
termed probabilistic point constraints (PPC) for implementing soft constraints. The PPC algorithm views constraints as stochastic measurements of the state of a static system.
The softness of a constraint is derived from the covariance of the "measurement." The resulting system of probabilistic equations is solved using the Kalman filter, a powerful
estimation tool in the theory of stochastic systems. We also describe a user interface using direct-manipulation devices for specifying and visualizing covariances in 2D and 3D.
The algorithm is suitable for any object represented as a parametric blend of control points, including most spline representations. The covariance of a constraint provides a
continuous transition from exact interpolation to controlled approximation of the constraint. The algorithm involves only linear operations and allows real-time interactive direct
manipulation of curves and surfaces on current workstations.
Suggested/Internal Citation Key
Rappoport:1994:IDO
#9:
Masatoshi Niizeki and Fujio Yamaguchi. Projectively Invariant Intersection Detections for Solid Modeling, ACM Transactions on Graphics, 13(3), pp. 277-299 (July
1994). ISSN 0730-0301.
Copyright
Copyright © 1994 Association for Computing Machinery
Abstract
An intersection detection method for solid modeling which is invariant under projective transformations is presented. We redefine the fundamental geometric figures necessary to
describe solid models and their dual figures in a homogeneous coordinate representation. Then we derive conditions, which are projectively invariant, for intersections between
these primitives. We will show that a geometric processor based on the 4 x 4 determinant method is applicable to a wide range of problems with little modification. This method
has applications in intersection detections of rational parametric curves and surfaces and hidden-line/surface removal algorithms.
Suggested/Internal Citation Key
Niizeki:1994:PII
#10:
Javier Sáchez-Reyes. Applications of the polynomial s-power basis in geometry processing, ACM Transactions on Graphics, 19(1), pp. 27-55 (January 2000). ISSN
0730-0301.
Keyword(s)
Hermite interpolation, Taylor series, degree reduction, geometry processing, offset curves and surfaces, power basis, s-power basis
Copyright
Copyright © 2000 Association for Computing Machinery
Abstract
We propose a unified methodology to tackle geometry processing operations admitting explicit algebraic expressions. This new approach is based on representing and
manipulating polynomials algebraically in a recently basis, the symmetric analogue of the power form (s-power basis for brevity), so called because it is associated with a
"Hermite two-point expansion" instead of a Taylor expansion. Given the expression of a polynomial in this basis over the unit interval u [epsilon][0, 1], degree reduction is
trivally obtained by truncation, which yields the He many terms as desired of the corresponding Hermite interpolant and build "s-power series," akin to Taylor series.
Applications include computing integral approximations of rational polynomials, or approximations of offset curves.
Suggested/Internal Citation Key
Sachez-Reyes:2000:AOT
#11:
Jianmin Zheng and Thomas W. Sederberg. Estimating tessellation parameter intervals for rational curves and surfaces, ACM Transactions on Graphics, 19(1), pp.
56-77 (January 2000). ISSN 0730-0301.
Keyword(s)
derivative bounds, flatness, projection distance, rational curves and surfaces, step size, tessellation
Copyright
Copyright © 2000 Association for Computing Machinery
Abstract
This paper presents a method for determining a priori a constant parameter interval for tessellating a rational curve or surface such that the deviation of the curve or surface from
its piecewise linear approximation is within a specified tolerance. The parameter interval is estimated based on information about second-order derivatives in the homogeneous
coordinates, instead of using affine coordinates directly. This new step size can be found with roughly the same amount of computation as the step size in Cheng [1992], though
it can be proven to always be larger than Cheng's step size. In fact, numerical experiments show the new step is typically orders of magnitude larger than the step size in Cheng
[1992]. Furthermore, for rational cubic and quartic curves, the new step size is generally twice as large as the step size found by computing bounds on the Bernstein polynomial
coefficients of the second derivatives function.
Suggested/Internal Citation Key
Zheng:2000:ETP
#12:
Insung Ihm and Bruce Naylor. Piecewise linear approximations of digitized space curves with applications, Scientific Visualization of Physical Phenomena (Proceedings of CG
International '91), pp. 545-569 (1991). Springer-Verlag. Edited by N. M. Patrikalakis.
Keyword(s)
piecewise linear approximation, digitized space curves, computational geometry, algebraic curves and surfaces, bsp tree
Suggested/Internal Citation Key
Ihm:1991:PLA
#13:
Gershon Elber. Metamorphosis of Free-form Curves and Surfaces, Computer Graphics International '95, (June 1995).
Suggested/Internal Citation Key
Elber:1995:MOF
#14:
H. Müller and R. Jaeschke. Adaptive Subdivision Curves and Surfaces, Computer Graphics International 1998, (June 1998, Hannover, Germany). IEEE Computer Society.
Copyright
Copyright © 1998 IEEE
Suggested/Internal Citation Key
Muller:1998:ASC
#15:
S. Takahashi. Geometric- and Parametric-Tolerance Constraints in Variational Design of Multiresolution Curves and Surfaces, Computer Graphics International 1998, (June
1998, Hannover, Germany). IEEE Computer Society.
Copyright
Copyright © 1998 IEEE
Suggested/Internal Citation Key
Takahashi:1998:GAP
#16:
Ryutarou Ohbuchi and Hiroshi Masuda and Masaki Aono. A Shape-Preserving Data Embedding Algorithm for NURBS Curves and Surfaces, Computer Graphics International
'99, (June 1999). IEEE CS Press . ISBN ISBN 0-7695-0185-0.
Suggested/Internal Citation Key
Ohbuchi:1999:ASD
#17:
H. G. Timmer. Alternative Representation for Parametric Cubic Curves and Surfaces, Computer-Aided Design, 12 (), pp. 25-28 (January 1980).
Keyword(s)
Algorithmic Aspects representation and surface representation
Suggested/Internal Citation Key
Timmer:1980:ARF
#18:
W. Boehm. Generating the Bézier Points of B-Spline Curves and Surfaces, Computer-Aided Design, 13 (), pp. 365-366 (November 1981).
Keyword(s)
B-spline
Suggested/Internal Citation Key
Boehm:1981:GTB
#19:
P. Bézier and S. Sioussiou. Semi-automatic system for defining free-form curves and surfaces, Computer Aided Design, 15 (2), pp. 65-72 (1983).
Suggested/Internal Citation Key
Bezier:1983:SSF
#20:
D. Ferguson. Construction of curves and surfaces using numerical optimization techniques, Computer Aided Design, 18 (1), pp. 15-21 (1986).
Suggested/Internal Citation Key
Ferguson:1986:COC
#21:
W. Boehm. Curvature Continuous Curves and Surfaces, Computer-Aided Design, 18 (2), pp. 105-106 (March 1986).
Keyword(s)
splines (mathematics), computational geometry, curvature continuous surfaces, cubic spline curves
Suggested/Internal Citation Key
Boehm:1986:CCC
#22:
Leslie Piegl. Representation of Rational Bézier Curves and Surfaces by Recursive Algorithms, Computer-Aided Design, 18 (7), pp. 361-366 (September 1986).
Suggested/Internal Citation Key
Piegl:1986:ROR
#23:
S. Abhyankar and C. Bajaj. Automatic parametrization of rational curves and surfaces II: cubics and cubicoids, Computer Aided Design, 19 (9), (1987).
Suggested/Internal Citation Key
Abhyankar:1987:APO
#24:
C. Lin. Generalized Bernstein-Bézier curves and surfaces, Computer-Aided Design, 20 (5), pp. 259-262 (1988).
Suggested/Internal Citation Key
Lin:1988:GBC
#25:
H. Hochfeld and M. Ahlers. Role of Bézier curves and surfaces in the Volkswagen CAD approach from 1967 to today, Computer-aided Design, 22 (9), pp.
598-608 (1990).
Suggested/Internal Citation Key
Hochfeld:1990:ROB
#26:
T. Goodman and H. Said. Properties of generalized Ball curves and surfaces, Computer Aided Design, 23 (8), pp. 554-560 (1991).
Suggested/Internal Citation Key
Goodman:1991:POG
#27:
L. Kocic. Modification of Bézier curves and surfaces by degree elevation technique, Computer Aided Design, 23 (10), pp. 692-699 (1991).
Suggested/Internal Citation Key
Kocic:1991:MOB
#28:
M. Kosters. Curvature-dependent parametrization of curves and surfaces, Computer Aided Design, 23 (8), pp. 569-578 (1991).
Suggested/Internal Citation Key
Kosters:1991:CPO
#29:
Richard D. Fuhr and Lwo Hsieh and Michael Kallay. Object-oriented paradigm for NURBS curve and surface design, Computer-aided Design, 27(2), pp.
95-100 (1995). Elsevier Science.
Keyword(s)
NURBS; curves and surfaces; algorithms
Abstract
The prevailing paradigm in the design of curve and surface algorithms could be described as being control-points oriented. The limitations of this approach are explored, and an
alternative paradigm is presented that combines object-oriented design with two well known approximation algorithms that are rarely mentioned in the cad literature. It provides a
powerful and versatile tool for constructing NURBS entities.
Suggested/Internal Citation Key
Fuhr:1995:OPF
#30:
Ardeshir Goshtasby. Geometric modelling using rational Gaussian curves and surfaces, Computer-aided Design, 27(5), pp. 363-375 (1995). Elsevier Science.
Keyword(s)
geometric modelling; rational Gaussian curves and surfaces; shapes
Abstract
A geometric modelling system based on rational Gaussian (RaG) curves and surfaces is introduced. The generation of simple geometric primitives such as lines, circles, and
ellipses with RaG curves, and the generation of planes, spheres, ellipsoids, cylinders, cones, and tori with RaG surfaces are discussed. The design of freeform closed,
half-closed, and open shapes using RaG surfaces is also considered. The control points of a RaG surface are not required to form a topologically rectangular grid, but, rather,
they can form an arbitrary grid.
Suggested/Internal Citation Key
Goshtasby:1995:GMU
#31:
Günther Greiner and Andreas Kolb and Ronald Pfeifle and Hans-Peter Seidel and Philipp Slusallek and Miguel Encarnação and Reinhard Klein. A platform for visualizing
curves and surfaces, Computer-aided Design, 27(7), pp. 559-566 (1995). Elsevier Science.
Keyword(s)
object-oriented graphics; surface interrogation; differential geometry
Abstract
Curves and surfaces have properties that are difficult to comprehend from a purely symbolic or numeric description. Therefore, visualization is an indispensable tool for
controlling the quality and for judging the aesthetic properties of these geometric objects. The same techniques can also be used advantageously for visually evaluating new
algorithms and mathematical schemes.The paper describes an object-oriented framework written in c++ that provides tools for the analysis and visualization of curves and
surfaces. The design of the class hierarchy is outlined, and specific applications (i.e. scattered data interpolation, blending surfaces, and differential geometry) that take
advantage of this platform are presented.
Suggested/Internal Citation Key
Greiner:1995:APF
#32:
Les Piegl and Wayne Tiller. Algorithm for degree reduction of B-spline curves, Computer-aided Design, 27(2), pp. 101-110 (1995). Elsevier Science.
Keyword(s)
B-splines; degree reduction; curves and surfaces
Abstract
An algorithmic approach to degree reduction of B-spline curves is presented. The method consists of the following steps: (a) decompose the B-spline curve into Bézier pieces on
the fly, (b) degree reduce each Bézier piece, and (c) remove the unnecessary knots. A complete algorithm and precise error control are provided.
Suggested/Internal Citation Key
Piegl:1995:AFD
#33:
Michael G. Wagner. Planar rational B-spline motions, Computer-aided Design, 27(2), pp. 129-137 (1995). Elsevier Science.
Keyword(s)
motions; nurbs curves; kinematic mapping
Abstract
Nonuniform rational B-spline (nurbs) curves and their associated techniques are of major importance in computer aided geometric design. The paper discusses planar rational
B-spline motions. These are planar motions in which all point paths are NURBS curves. Such motions are connected with a linear control structure, which can be used to apply
algorithms developed for the design of curves and surfaces directly to the design of planar motions.The first part of the paper gives a brief introduction to plane kinematics
and the theory of kinematic mappings. Rational motions and the application of the corresponding control structures are discussed in detail. The second part of the paper presents
a C2 interpolation scheme with rational motions of degree 4, which is the minimum degree for motions which have positions with vanishing angular velocity.
Suggested/Internal Citation Key
Wagner:1995:PRB
#34:
Ma Weiyin and J. P. Kruth. Parameterization of randomly measured points for least squares fitting of B-spline curves and surfaces, Computer-aided Design, 27(9), pp.
663-675 (1995). Elsevier Science.
Keyword(s)
parameterization; interpolation; B-spline surfaces
Abstract
The paper presents a simple technique to assign parameter values to randomly measured points for the least squares fitting of B-spline surfaces. The parameterization is realized
by projecting the measured points to a base surface. The parameters of the projected points are then used as the parameters of the measured points. The base surface is in fact a
first approximation of the final fitted surface, and it can usually be created from some approximate boundary information in the form of either points or curves. A similar
technique can also be used for B-spline curve fitting.
Suggested/Internal Citation Key
Weiyin:1995:POR
#35:
Xiuzi Ye. Generating Béier points for curves and surfaces from boundary information, Computer-aided Design, 27(12), pp. 875-885 (1995). Elsevier Science.
Keyword(s)
product data exchange; change of basis; Bézier curves; Bézier patches; Hermite curves; Coons-Hermite Cartesian sum patches; Coons-Boolean sum patches
Abstract
This paper presents efficient methods for directly generating Bézier points of curves and surfaces explicitly from the given compatible arbitrary order boundary information
of Hermite curves, Coons-Hermite Cartesian sum patches and Coons-Boolean sum patches. The explicit expressions for the generalized Hermite functions are also developed.
Furthermore, a method for determining the twist control points and higher level sets of interior control points from their boundary and lower level sets of control points by using
the Coons-Boolean sum schema presented. Many interesting and useful examples are also given in this paper.
Suggested/Internal Citation Key
Ye:1995:GBP
#36:
Les Piegl and Wayne Tiller. Algorithm for approximate NURBS skinning, Computer-aided Design, 28(9), pp. 699-706 (1996). Elsevier Science.
Keyword(s)
nurbs; surface skinning; curves and surfaces; algorithms
Abstract
An algorithm for approximate skinning through cross-sectional nurbs curves is presented. The method eliminates the problem of dealing with huge amounts of control points
obtained during the curve compatability process. It also allows the designer to specify large numbers of cross-sections and approximately fit a smooth surface to these curves to
any given tolerance. Depending on the tolerances used, up to 99% of the control points can be eliminated.
Suggested/Internal Citation Key
Piegl:1996:AFA
#37:
Les Piegl and Wayne Tiller. Symbolic operators for NURBS, Computer-aided Design, 29(5), pp. 361-368 (1997). Elsevier Science.
Keyword(s)
NURBS; symbolic operators; geometric algorithms
Abstract
Symbolic operators for NURBS curves and surfaces are presented in this paper. The operators are used to compute NURBS entities by performing algebraic operations
using NURBS curves and surfaces as variables. Dot and cross products, sum/difference and derivative operators are presented. An application to construct ruled surfaces to
rational rail curves is also included.
Suggested/Internal Citation Key
Piegl:1997:SOF
#38:
Junji Ishida. The general B-spline interpolation method and its application to the modification of curves and surfaces, Computer-aided Design, 29(11), pp.
779-790 (1997). Elsevier Science.
Keyword(s)
modification of curves and surfaces; Karlin-Ziegler theorem; Gordon surface; abbreviated headline; deformation of geometry
Abstract
Direct manipulation of B-spline control points has been used for modifying B-spline curves and surfaces. But, designers usually wish to modify shapes in more direct ways
in practical designing situations, such as moving a point on a curve to some desirable location or modifying a tangent vector at some point on a curve into another direction etc.
The author will propose a method that enables arbitrary and direct modification of curves by constructing a displacement function. Moreover, a systematic B-spline interpolation
method which has enough generality for practical use will be proposed. The method is also available for surfaces and some interesting applications will be shown.
Suggested/Internal Citation Key
Ishida:1997:TGB
#39:
S. L. Abrams and W. Cho and C.-Y. Hu and T. Maekawa and N. M. Patrikalakis and E. C. Sherbrooke and X. Ye. Efficient and reliable methods for rounded-interval
arithmetic, Computer-aided Design, 30(8), pp. 657-665 (1998). Elsevier Science.
Keyword(s)
binary representation; denormalized number; IEEE Std 754-1985; rounded-interval arithmetic; unit-in-the-last-place
Abstract
We present an efficient and reliable method for computing the unit-in-the-last-place (ulp) of a double-precision floating-point number, taking advantage of the standard binary
representation for floating-point numbers defined by IEEE Std 754-1985. The ulp is necessary to perform software rounding for robust rounded-interval arithmetic (RIA)
operations. Hardware rounding, using two of the standard rounding modes defined by IEEE-754, may be more efficient. RIA has been used to produce robust software systems
for the solution of systems of nonlinear equations, interrogation of geometric and differential properties of curves and surfaces, curve and surface intersections, and solid
modeling.
Suggested/Internal Citation Key
Abrams:1998:EAR
#40:
Takashi Maekawa. An overview of offset curves and surfaces, Computer-aided Design, 31(3), pp. 165-173 (1999). Elsevier Science.
Keyword(s)
Offset curves; Offset surfaces; Self-intersections; Pythagorean hodographs; Geodesics offsets; General offsets
Copyright
Copyright © 1999 Elsevier Science
Abstract
A literature survey on offset curves and surfaces up to 1992 was carried out by Pham (Pham B, Offset curves and surfaces: a brief survey. Computer Aided Design
1992; 24(4): 223-229). The objective of this article is to overview the literature after 1992 and those which were not cited in aforementioned paper. The article focuses on five
active areas of research on offsets: (1) representing exact offsets in Bézier/B-spline format, (2) approximations, (3) self-intersections, (4) geodesic offsets and (5) general
offsets.
Suggested/Internal Citation Key
Maekawa:1999:AOO
#41:
K. C. Hui. Shape blending of curves and surfaces with geometric continuity, Computer-Aided Design, 31(13), pp. 819-828 (November 1999). ISSN 0010-4485.
Keyword(s)
Geometric continuity, Morphing, Parametric curves, Parametric surfaces, Ruled surfaces, Bézier patches
Copyright
Copyright © 1999 Elsevier Science
Abstract
Linear interpolation between G1 piecewise continuous curves may result in geometrically non-continuous curves. This affects the continuity of the shapes created in a
shape-blending process. Similar effects also affect the continuity of the ruled surfaces constructed with G1 continuous curves. This paper presents anapproach for maintaining
G1 continuity of the blended curves by adjusting the positions of the junction points of the curve segments. Criterion for G1 continuity of ruled surfaces are studied and the
sufficient conditions for G1 continuity are identified. G1 continuity of composite surfaces in a shape-blending process is also studied. An approach is proposed to maintain
the G1 continuity of Bézier surfaces pairs in a shape-blending process by adjusting the control points along the common boundary of the resulting surface-pair. This is
extended for retaining G1 continuity of shape-blended Bézier surfaces sharing a common corner.
Suggested/Internal Citation Key
Hui:1999:SBO
#42:
J. Hoschek. Smoothing of curves and surfaces, Computer Aided Geometric Design, 2 (1-3), pp. 97-105 (1985).
Suggested/Internal Citation Key
Hoschek:1985:SOC
#43:
W. Boehm. Curvature continuous curves and surfaces, Computer Aided Geometric Design, 2 (2), pp. 313-323 (1985).
Suggested/Internal Citation Key
Boehm:1985:CCC
#44:
T. Lyche and V. Morken. Knot removal for parametric B-spline curves and surfaces, Computer Aided Geometric Design, 4 (3), pp. 217-230 (1987).
Suggested/Internal Citation Key
Lyche:1987:KRF
#45:
Michael A. Lachance. Chebyshev economization for parametric surfaces, Computer Aided Geometric Design, 5 (3), pp. 195-208 (September 1988).
Keyword(s)
parametric polynomial curves and surfaces, constrained chebyshev polynomials, contrained chebyshev economization, remez algorithm
Suggested/Internal Citation Key
Lachance:1988:CEF
#46:
R. Farouki and V. Rajan. On the numerical condition of algebraic curves and surfaces - 1. Implicit equations, Computer Aided Geometric Design, 5 (4), pp.
215-252 (1988).
Suggested/Internal Citation Key
Farouki:1988:OTN
#47:
S. Abhyankar and C. Bajaj. Automatic parametrization of rational curves and surfaces III: Algebraic plane curves, Computer Aided Geometric Design, 5 (4), pp.
309-322 (1988).
Suggested/Internal Citation Key
Abhyankar:1988:APO
#48:
M. Daniel and J. C. Daubisse. The numerical problem of using Bézier curves and surfaces in the power basis, Computer Aided Geometric Design, 6 (2), pp.
121-128 (1989).
Keyword(s)
bezier modeling, matrix conditioning, number of significant digits
Suggested/Internal Citation Key
Daniel:1989:TNP
#49:
N. Dyn and D. Levin and C. Micchelli. Using parameters to increase smoothness of curves and surfaces generated by subdivision, Computer Aided Geometric Design, 7
(1-4), pp. 129-140 (1990).
Suggested/Internal Citation Key
Dyn:1990:UPT
#50:
T. Jensen and C. Petersen and M. Watkins. Practical curves and surfaces for a geometric modeler, Computer Aided Geometric Design, 8 (5), pp. 357-370 (1991).
Suggested/Internal Citation Key
Jensen:1991:PCA
#51:
B. Sarkar and C. Menq. Parameter optimization in approximating curves and surfaces to measurement data, Computer Aided Geometric Design, 8 (4), pp.
267-290 (1991).
Suggested/Internal Citation Key
Sarkar:1991:POI
#52:
R. Dietz and J. Hoschek and B. Jüttler. An algebraic approach to curves and surfaces on the sphere and on other quadrics, Computer Aided Geometric Design, 10 (3), pp.
211-230 (August 1993).
Suggested/Internal Citation Key
Dietz:1993:AAA
#53:
Marie-Laurence Mazure. Geometric contact for curves and surfaces, Computer Aided Geometric Design, 11(2), pp. 177-195 (1994). Elsevier Science. ISSN 0167-8396.
Keyword(s)
Parametric curves; Parametric surfaces; Geometric contact; Frénet contact; Connection matrices; Frénet frame; Geometric invariants
Abstract
The notion of Frénet-contact of order p (Fp) for parametric curves in Rd is defined in a new geometrical way which allows its extension to the case of surfaces. This notion is
compared to geometric continuity of order p (Gp).
Suggested/Internal Citation Key
Mazure:1994:GCF
#54:
C. Carstensen and G. Mühlbach and G. Schmidt. De Casteljau's algorithm is an extrapolation method, Computer Aided Geometric Design, 12(4), pp.
371-380 (1995). Elsevier Science. ISSN 0167-8396.
Keyword(s)
Recurrence scheme; de Casteljau's algorithm; Bernstein polynomials; Extrapolation algorithms; E-algorithm; GNA-algorithm
Abstract
One of the most important recursive schemes in CAGD is de Casteljau's algorithm for the evaluation of Bézier curves and surfaces. Within the theory of triangular recursive
schemes we discuss the De Casteljau's algorithm as a particular case, i.e. we prove that it is identical to the E-algorithm (or GNA-algorithm) in a particular frame. This result is
of theoretical interest since it leads to some classification of recurrence relations in CAGD. Furthermore, it may be regarded as a model example how to obtain known and
possibly new recursive schemes in CAGD as examples of the theory of general extrapolation algorithms.
Suggested/Internal Citation Key
Carstensen:1995:DCA
#55:
Sean M. Gelston and Debasish Dutta. Boundary surface recovery from skeleton curves and surfaces, Computer Aided Geometric Design, 12(1), pp.
27-51 (1995). Elsevier Science. ISSN 0167-8396.
Abstract
Medial axis transforms, or skeletons, have many applications in computer aided geometric design and analysis. Construction of skeletons is an active area of research. We
consider the inverse problem, that of recovering boundary surfaces from given skeleton elements. The skeleton of any 3D object will, in general, consist of curves and
surfaces. In this paper, we first outline a method for reconstructing boundary surfaces corresponding to skeletal curves, and then extend the method for reconstruction of
boundary surfaces corresponding to skeletal surfaces. Implemented examples for both curves and surfaces are included.
Suggested/Internal Citation Key
Gelston:1995:BSR
#56:
Helmut Pottmann. Rational curves and surfaces with rational offsets, Computer Aided Geometric Design, 12(2), pp. 175-192 (1995). Elsevier Science. ISSN 0167-8396.
Keyword(s)
Rational curve; Rational surface; Offset curve; Offset surface; Rational Bézier representation; Dual Bézier curves and surfaces; Spherical Bézier patch; Isophote
Abstract
Given a rational algebraic surface in the rational parametric representation s->(u,v) with unit normal vectors n->(u,v)=(s->u ? s->v)/ || s->u ?s->v || , the offset surface
at distance d is s->d(u,v)=s->(u,v)+dn->(u,v) . This is in general not a rational representation, since || s->u ? s->v || is in general not rational. In this paper, we present an
explicit representation of all rational surfaces with a continuous set of rational offsets s->d(u,v). The analogous question is solved for curves, which is an extension of
Farouki's Pythagorean hodograph curves to the rationals. Additionally, we describe all rational curves c->(t) whose arc length parameter s(t) is a rational function of t . Offsets
arise in the mathematical description of milling processes and in the representation of thick plates, such that the presented curves and surfaces possess a very attractive property
for practical use.
Suggested/Internal Citation Key
Pottmann:1995:RCA
#57:
Takafumi Saito and Guo-Jin Wang and Thomas W. Sederberg. Hodographs and normals of rational curves and surfaces, Computer Aided Geometric Design, 12(4), pp.
417-430 (1995). Elsevier Science. ISSN 0167-8396.
Keyword(s)
Rational curves; Rational surfaces; Hodographs; Normal vectors
Abstract
Derivatives and normals of rational Bézier curves and surface patches are discussed. A non-uniformly scaled hodograph of a degree m xn tensor-product rational surface, which
provides correct derivative direction but not magnitude, can be written as a degree (2m - 2) ? 2n or 2m ? (2n - 2) vector function in polynomial Bézier form. Likewise, the
scaled normal direction is degree (3m - 2) ?(3n - 2). Efficient methods are developed for bounding these directions and the derivative magnitude.
Suggested/Internal Citation Key
Saito:1995:HAN
#58:
Ahmed Khamayseh and Bernd Hamann. Elliptic grid generation using NURBS surfaces, Computer Aided Geometric Design, 13(4), pp. 369-386 (1996). Elsevier
Science. ISSN 0167-8396.
Keyword(s)
Algebraic grid generation; Approximation; Elliptic grid generation; NURBS curve; NURBS surface; Partial differential equations; Transfinite interpolation
Abstract
Recently, there has been a move towards NURBS-based grid generation systems, where the original geometry is given as analytically defined NURBS surfaces. The process of
surface grid generation is the computation of an algebraic grid based on the NURBS surface definition and the computation of an elliptic grid based on the algebraic grid. The
NURBS format provides a common mathematical representation for both standard analytic shapes and free-form curves and surfaces. The derivatives of the physical
coordinates with respect to the parametric coordinates can be evaluated directly. An improved elliptic surface grid generation method for NURBS surfaces is presented. New
techniques for computing the control functions and imposing boundary orthogonality are developed.
Suggested/Internal Citation Key
Khamayseh:1996:EGG
#59:
Kouichi Konno and Hiroaki Chiyokura. An approach of designing and controlling free-form surfaces by using NURBS boundary Gregory patches, Computer Aided Geometric
Design, 13(9), pp. 825-849 (1996). Elsevier Science. ISSN 0167-8396.
Keyword(s)
Curve mesh; Surface interpolation; Gregory patch; NURBS; NURBS boundary Gregory patch
Abstract
Designers require a means of designing complex free-form surfaces easily and intuitively. One general approach to designing such surfaces is to first define a curve mesh
consisting of characteristic lines, such as cross sections and boundary curves, then to interpolate the curve mesh using free-form surfaces. NURBS surfaces are widely used but
make the interpolation of an irregular curve mesh difficult. This has been a major limiting constraint on designers. In this paper, we propose a new surface representation that
enables the smooth interpolation of an irregular curve mesh with NURBS curves and surfaces.
Suggested/Internal Citation Key
Konno:1996:AAO
#60:
Jiwen Zhang. C-curves: An extension of cubic curves, Computer Aided Geometric Design, 13(3), pp. 199-217 (1996). Elsevier Science. ISSN 0167-8396.
Keyword(s)
Cubic C-Ferguson curve; Bézier curve; Uniform B-spline; C-Ferguson curve; C-Bézier curve; C-B-spline; NURBS
Abstract
A linearly parametrized set of curves, named C-curves, is suggested with basis sin t, cos t, t, and 1. C-curves are an extension of cubic curves, they depend on a parameter
>alpha<>0, and their limiting case for >alpha<->0 is a cubic curve. They can deal with free form curves and surfaces, and provide exact reproduction of circles and cylinders.
So, they could be used to unify the representation and processing of both free and normal form curves and surfaces in engineering.
Suggested/Internal Citation Key
Zhang:1996:CAE
#61:
C-curves: An extension of cubic curves,
#62:
Michael S. Floater. An O(h2n) Hermite approximation for conic sections, Computer Aided Geometric Design, 14(2), pp. 135-151 (1997). Elsevier Science. ISSN
0167-8396.
Keyword(s)
High order approximation; Conic sections; Splines
Abstract
Given a segment of a conic section in the form of a rational quadratic Bézier curve and any positive odd integer n , a geometric Hermite interpolant, with 2n contacts, counting
multiplicity, is presented. This leads to a Gn-1 spline approximation having an approximation order of O(h2n) . A bound on the Hausdorff error of the Hermite interpolant
is provided. Both the interpolation and error bound are extended to an important subclass of rational biquadratic Bézier surfaces. For low n , the approximation provides a
method for converting the so-called analytic curves and surfaces used in CAGD to polynomial spline form with very small error.
Suggested/Internal Citation Key
Floater:1997:AH
#63:
Johannes Wallner and Helmut Pottmann. Rational blending surfaces between quadrics, Computer Aided Geometric Design, 14(5), pp. 407-419 (1997). Elsevier
Science. ISSN 0167-8396.
Keyword(s)
Kinematic mapping; Line geometry; NURBS surface; Blending surface
Abstract
Using tools from classical line geometry and the theory of kinematic mappings, it is possible to define an intrinsic control structure for NURBS curves and surfaces on the
sphere, the cylinder and on any projectively equivalent quadratic surface. These methods are further used to construct exact C1 blends between these surfaces, such that
interactive design of trim lines and surface tension is possible. The lowest possible degree of a blend that can be achieved with this method is (4,3) .
Suggested/Internal Citation Key
Wallner:1997:RBS
#64:
Dinesh Manocha and Shankar Krishnan. Algebraic pruning: a fast technique for curve and surface intersection, Computer Aided Geometric Design, 14(9), pp.
823-845 (1997). Elsevier Science. ISSN 0167-8396.
Keyword(s)
Intersection; Curves; Surfaces; Ray-tracing; Resultants; Eigendecomposition; Solid modeling
Abstract
Computing the intersection of parametric and algebraic curves and surfaces is a fundamental problem in computer graphics and geometric modeling. This problem has been
extensively studied in the literature and different techniques based on subdivision, interval analysis and algebraic formulation are known. For low degree curves and surfaces
algebraic methods are considered to be the fastest, whereas techniques based on subdivision and Bézier clipping perform better for higher degree intersections. In this paper, we
introduce a new technique of algebraic pruning based on the algebraic approaches and eigenvalue formulation of the problem. The resulting algorithm corresponds to computing
only selected eigenvalues in the domain of intersection. This is based on matrix formulation of the intersection problem, power iterations and geometric properties of Bézier
curves and surfaces. The algorithm prunes the domain and converges to the solutions rapidly. It has been applied to intersection of parametric and algebraic curves, ray
tracing and curve-surface intersections. The resulting algorithm compares favorably with earlier methods in terms of performance and accuracy.
Suggested/Internal Citation Key
Manocha:1997:APA
#65:
L.-E. Andersson and T. J. Peters and N. F. Stewart. Selfintersection of composite curves and surfaces, Computer Aided Geometric Design, 15(5), pp.
507-527 (1998). Elsevier Science. ISSN 0167-8396.
Abstract
This paper provides computationally tractable conditions to determine whether a composite spline curve or patch selfintersects, according to a definition that includes the
important limiting cases of cusps, singularities, and tangential intersections of adjacent components. These results follow upon our exposition of necessary and sufficient
conditions to preclude such selfintersections. The paper includes a numerical example illustrating the results, and discusses an important application, namely, guaranteeing that a
finite curvilinear simplicial complex in R3 , made up of properly-joined parametric patches,
Suggested/Internal Citation Key
Andersson:1998:SOC
#66:
Yong-Ming Li and Xiao-Ying Zhang. Basis conversion among Bézier, Tchebyshev and Legendre, Computer Aided Geometric Design, 15(6), pp. 637-642 (1998). Elsevier
Science. ISSN 0167-8396.
Keyword(s)
Basis conversion; Bézier; Tchebyshev; Legendre polynomials
Abstract
Bézier representation of curves and surfaces has been a standard in most CAD/CAM systems. The conversion between Bézier, Tchebyshev, and Legendre representation of
polynomial curves and surfaces is often desired when an approximation procedure is involved. In this paper, we present a simple and numerically stable method for basis
conversion of Bézier, Tchebyshev, and Legendre polynomials.
Suggested/Internal Citation Key
Li:1998:BCA
#67:
Martin Peternell and Helmut Pottmann. A Laguerre geometric approach to rational offsets, Computer Aided Geometric Design, 15(3), pp. 223-249 (1998). Elsevier
Science. ISSN 0167-8396.
Keyword(s)
Laguerre geometry; NC milling; geometrical optics; rational curve; rational surface; offset; rational offset; Pythagorean-hodograph curve; principal patch; principal curvature line
Abstract
Laguerre geometry provides a simple approach to the design of rational curves and surfaces with rational offsets. These so-called PH curves and PN surfaces can be
constructed from arbitrary rational curves or surfaces with help of a geometric transformation which describes a change between two models of Laguerre geometry. Closely
related to that is their optical interpretation as anticaustics of arbitrary rational curves/surfaces for parallel illumination. A theorem on rational parametrizations for envelopes of
natural quadrics leads to algorithms for the computation of rational parametrizations of surfaces; those include canal surfaces with rational spine curve and rational radius
function, offsets of rational ruled surfaces or quadrics, and surfaces generated by peripheral milling with a cylindrical or conical cutter. Laguerre geometry is also useful for the
construction of PN surfaces with rational principal curvature lines. New families of such principal PN surfaces are determined.
Suggested/Internal Citation Key
Peternell:1998:ALG
#68:
Les A. Piegl and Wayne Tiller. Computing the derivative of NURBS with respect to a knot, Computer Aided Geometric Design, 15(9), pp. 925-934 (1998). Elsevier
Science. ISSN 0167-8396.
Keyword(s)
B-splines; Differentiation; Symbolic operators
Abstract
Algorithms for computing the derivative of NURBS with respect to a knot are presented. Rational and nonrational curves and surfaces as well as basis functions are
differentiated with respect to a knot. The derivative entities are computed by control point or basis function differencing divided by appropriate knot spans.
Suggested/Internal Citation Key
Piegl:1998:CTD
#69:
Rida T. Farouki and Yi-Feng Tsai and Guo-Feng Yuan. Contour machining of free-form surfaces with real-time PH curve CNC interpolators, Computer Aided Geometric
Design, 16(1), pp. 61-76 (1999). Elsevier Science. ISSN 0167-8396.
Keyword(s)
machining; Pythagorean-hodograph curves; Free-form surfaces; Surface contours; CNC interpolators; Feedrates; Offset curves and surfaces
Abstract
Two strategies for contour milling of free-form surfaces, using real-time CNC interpolators for Pythagorean-hodograph (PH) curves, are described. The first method, applicable
to convex surfaces, employs a flat-end mill and approximates the surface section curves by planar PH quintics. The second approach, which employs a ball-end mill,
approximates the tool-center trajectory by quintic PH space curves, and can accommodate nonconvex surfaces by choosing a sufficiently small tool radius. Both schemes
generate compact part programs, in which numerous short linear/circular G code motions are replaced by fewer analytic path segments, and eliminate the need for explicit offset
curve or surface representations to compensate for the tool radius. The surface sectioning and PH curve approximation algorithms required by these methods are presented, with
appropriate tolerance analyses, and preliminary results from machining experiments performed on an open-architecture 3-axis CNC mill are described.
Suggested/Internal Citation Key
Farouki:1999:CMO
#70:
Erich Hartmann. On the curvature of curves and surfaces defined by normalforms, Computer Aided Geometric Design, 16(5), pp. 355-376 (1999). Elsevier
Science. ISSN 0167-8396.
Keyword(s)
Normalform; Hessian matrix; Curvature; Normal curvature; Bisector; Gn-blending; G2-continuity; Umbilic points; Isophote; Curvature line; Feature line; Ridge; Ravine;
Intersection curve; Foot point
Abstract
The normalform h=0 of a curve (surface) is a generalization of the Hesse normalform of a line in R2 (plane in R3). It was introduced and applied to curve and surface design
in recent papers. For determining the curvature of a curve (surface) defined via normalforms it is necessary to have formulas for the second derivatives of the normalform
function h depending on the unit normal and the normal curvatures of three tangential directions of the surface. These are derived and applied to visualization of the curvature of
bisectors and blending curves, isophotes, curvature lines, feature lines and intersection curves of surfaces. The idea of the normalform is an appropriate tool for proving
theoretical statements, too. As an example a simple proof of the Linkage Curve Theorem is given.
Suggested/Internal Citation Key
Hartmann:1999:OTC
#71:
Ming Zhang and Eng-Wee Chionh and Ronald N. Goldman. On a relationship between the moving line and moving conic coefficient matrices, Computer Aided Geometric
Design, 16(6), pp. 517-527 (1999). Elsevier Science. ISSN 0167-8396.
Keyword(s)
Implicitization; Moving line; Moving conic
Abstract
The method of moving curves and moving surfaces is a new, effective tool for implicitizing rational curves and surfaces. Here we investigate a relationship between the
moving line coefficient matrix and the moving conic coefficient matrix for rational curves. Based on this relationship, we present a new proof that the method of moving conics
always produces the implicit equation of a rational curve when there are no low degree moving lines that follow the curve.
Suggested/Internal Citation Key
Zhang:1999:OAR
#72:
Erich Hartmann. Numerical parameterization of curves and surfaces, Computer Aided Geometric Design, 17(3), pp. 251-266 (March 2000). ISSN 0167-8396.
Keyword(s)
Parameterization, Implicit curve, Implicit surface, Normalform, Mesh generation, Curvature, Foot point, Intersection curve
Copyright
Copyright © 2000 Elsevier Science
Abstract
A method for parameterizing nearly arbitrary implicit plane/space curves and surfaces is introduced. The parameterizations are of class Cn-1 if the given curves/surfaces are
of class Cn. The computation of points and derivatives is performed numerically. These parameterizations can be used for controlled determination of points on curves and
surfaces and for the application of developed techniques for parametric curves and surfaces (mesh generation, texture mapping, curve integrals, surface integrals ...) to
implicit curves and surfaces. The idea of the normalform of a curve/surface introduced in recent papers makes it possible to apply the numerical parameterization to nearly
arbitrary curves and surfaces.
Suggested/Internal Citation Key
Hartmann:2000:NPO
#73:
Géraldine Morin and Ron Goldman. A subdivision scheme for Poisson curves and surfaces, Computer Aided Geometric Design, 17(9), pp. 813-833 (October
2000). ISSN 0167-8396.
Keyword(s)
Analytic function, Bézier curve, de Casteljau algorithm, Poisson distribution, Stationary subdivision
Copyright
Copyright © 2000 Elsevier Science
Abstract
The de Casteljau evaluation algorithm applied to a finite sequence of control points defines a Bézier curve. This evaluation procedure also generates a subdivision algorithm and
the limit of the subdivision process is this same Bézier curve. Extending the de Casteljau subdivision algorithm to an infinite sequence of control points defines a new family of
curves. Here, limits of this stationary non-uniform subdivision process are shown to be equivalent to curves whose control points are the original data points and whose
blending functions are given by the Poisson distribution. Thus this approach generalizes standard subdivision techniques from polynomials to arbitrary analytic functions.
Extensions of this new subdivision scheme from curves to tensor product surfaces are also discussed.
Suggested/Internal Citation Key
Morin:2000:ASS
#74:
R. Kazinnik and G. Elber. Orthogonal Decomposition of Non-Uniform Bspline Spaces using Wavelets, Computer Graphics Forum, 16(3), pp. 27-38 (August
1997). Blackwell Publishers. Edited by Dieter Fellner and L. Szirmay-Kalos. ISSN 1067-7055.
Abstract
We take advantage of ideas of an orthogonal wavelet complement to produce multiresolution orthogonal decomposition of nonuniform Bspline (NUB) spaces. The editing of
NUB curves and surfaces can be handled at different levels of resolutions. Applying Multiresolution decomposition to possibly C1 discontintious surfaces, one can preserve
the general shape on one hand and local features on the other of the free-form models, including geometric discontinuities. The Multiresolution decomposition of the NUB tensor
product surface is computed via the symbolic computation of innerproducts of Bspline basis functions. To find a closed form representationfor the innerproduct of the Bspline
basis functions, an equivalent interpolation problem is solved. As an example for the strength of the Multiresolution decomposition, a tool demonstrating the Multiresolittion
editing capabilities of NUB surfaces was developed and is presented as part of this work, allowing interactive 3D editing of NUB free-form surfaces. Proceedings of
Eurographics '97.
Suggested/Internal Citation Key
Kazinnik:1997:ODO
#75:
S. Kuriyama and K. Tachibana. Polyhedral Surface Modeling with a Diffusion System, Computer Graphics Forum, 16(3), pp. 39-46 (August 1997). Blackwell
Publishers. Edited by Dieter Fellner and L. Szirmay-Kalos. ISSN 1067-7055.
Keyword(s)
Curves and Surfaces, Diffusion Systems, Polyhedral Surfaces, Tension Controls
Abstract
This paper presents a method of generating polyhedral surfaces by using a diffusion system that calculates the positional and normal vectors on their vertices. The system
generates smooth shapes that satisfy the minimum norm property, and can be extended to imitate the shape controls of curvature continuous surfaces with bias and tension
parameters. The shape of a surface is determined by the stable state of nonlinear and local calculations between vertices, and is easily controlled by adding constraints on
arbitrary vertices. Such bottom-up calculation of surfaces enhances flexibility in the interactive design of complicated free-form shapes. Proceedings of Eurographics '97.
Suggested/Internal Citation Key
Kuriyama:1997:PSM
#76:
F. W. B. Li and R. W. H. Lau and M. Green. Interactive Rendering of Deforming NURBS Surfaces, Computer Graphics Forum, 16(3), pp. 47-56 (August 1997). Blackwell
Publishers. Edited by Dieter Fellner and L. Szirmay-Kalos. ISSN 1067-7055.
Abstract
Non-uniform rational B-splines (NURBS) has been widely accepted as a standard tool for geometry representation and design. Its rich geometric properties allow it to represent
both analytic shapes and free-form curves and surfaces precisely. Moreover, a set of tools is available for shape modification or more implicitly, object deformation. Existing
NURBS rendering methods include de Boor algorithm, Oslo algorithm, Shantz's adaptive forward differencing algorithm and Silbermann's high speed implementation of
NURBS. However, these methods consider only speeding up the rendering process of individual frames. Recently, Kumar et al. proposed an incremental method for rendering
NURBS surfaces, but it is still limited to static surfaces. In real-time applications such as virtual reality, interactive display is needed If a virtual environment contains a lot of
deforming objects, these methods cannot provide a good solution. In this paper we propose an efficient method for interactive rendering of deformable objects by maintaining a
polygon model of each deforming NURBS surface and adaptively refining the resolution of the polygon model. We also took at how this method may be applied to
multi-resolution modelling. Proceedings of Eurographics '97.
Suggested/Internal Citation Key
Li:1997:IRO
#77:
Cindy Grimm and Matthew Ayers. A Framework for Synchronized Editing of Multiple Curve Representations, Computer Graphics Forum, 17(3), pp.
31-40 (1998). Blackwell Publishers. Edited by N. Ferreira and M. Göbel. ISSN 1067-7055.
Keyword(s)
direct manipulation, interface issues, curve manipulation
Abstract
Editing curves and surfaces is difficult in part because their mathematical representations rarely correspond to most people's idea of a curve or surface. The implementation
(and hence, behavior) of most manipulation tools is intertwined with a particular curve or surface representation; this can make reimplementing the tool with a different
representation problematic. A system using a single representation must therefore either limit the types of tools available or convert existing tools to work on the system's
representation.
In this paper we present a framework for editing curves or surfaces which supports multiple representations and ensures that they stay synchronized. As a proof of concept, we
have created a curve editor which contains several tools each of which manipulate one of three different curve representations: polylines, NURBS, and multi-resolution
B-splines.
Suggested/Internal Citation Key
Grimm:1998:AFF
#78:
J. M. Zheng and K. W. Chan and I. Gibson. A New Approach for Direct Manipulation of Free-Form Curve, Computer Graphics Forum, 17(3), pp.
327-334 (1998). Blackwell Publishers. Edited by N. Ferreira and M. Göbel. ISSN 1067-7055.
Abstract
There is an increasing demand for more intuitive methods for creating and modifying free-form curves and surfaces in CAD modeling systems. The methods should be
based not only on the change of the mathematical paraameters, such as control points, knots, and weights, but also on the user's specified constraints and shapes. This paper
presents a new approach for directly manipulating the shape of a free-form curve, leading to a better control of the curve deformation and a more intuitive CAD modeling
interface. The user's intended deformation of a curve is automatically converted into the modification of the corresponding NURBS control points and knot sequence of the
curve. The algorithm for this approach includes curve elevation, knot refinement, control point repositioning and knot removal. Several examples shown in this paper
demonstrate that the proposed method can be used to deform a NURBS curve into the desired shape. Currently, the algorithm concentrates on the purely geometric
consideration. Further work will include the effect of material properties.
Suggested/Internal Citation Key
Zheng:1998:ANA
#79:
Faramarz F. Samavati and Richard M. Bartels. Multiresolution Curve and Surface Representation: Reversing Subdivision Rules by Least-Squares Data Fitting, Computer
Graphics Forum, 18(2), pp. 97-119 (June 1999). Blackwell Publishers. ISSN 1067-7055.
Abstract
This work explores how three techniques for defining and representing curves and surfaces can be related efficiently. The techniques are subdivision, least-squares data
fitting, and wavelets. We show how least-squares data fitting can be used to "reverse" a subdivision rule, how this reversal is related to wavelets, how this relationship can
provide a multilevel representation, and how the decomposition/reconstruction process can be carried out in linear time and space through the use of a matrix factorization.
Some insights that this work brings forth are that the inner product used in a multiresolution analysis influences the support of a wavelet, that wavelets can be constructed by
straightforward matrix observations, and that matrix partitioning and factorization can provide alternatives to inverses or duals for building efficient decomposition and
reconstruction processes. We illustrate our findings using an example curve, grey-scale image, and tensor-product surface.
Suggested/Internal Citation Key
Samavati:1999:MCA
#80:
M. Fontana and F. Giannini and M. Meirana. A Free Form Feature Taxonomy, Computer Graphics Forum, 18(3), pp. 107-118 (September 1999). Blackwell
Publishers. ISSN 1067-7055.
Abstract
In this paper the notion of free form feature for aesthetic design is presented. The design of industrial products constituted by free form surfaces is done by using CAD systems
representing curves and surfaces by means of NURBS functions, which are usually defined by low level entities that are not intuitive and require some knowledge of the
mathematical language. Similarly to the feature-based approach adopted by CAD systems for classical mechanical design, a set of high level modelling entities which provides
commonly performed shape modifications has been identified. Particularly, the paper suggests a classification of the so-called detail features for an aesthetic and/or functional
characterization of predefined free form surfaces. Feature types are formally described by means of an analytical definition of the surface modification through deformation and
elimination laws. A topological classification is then given according to the application domain of such laws. A further sub-classification of morphological types is then
suggested according to geometric properties of weak convexity and concavity for the resulting modified shape, leading to a taxonomy of simple free form features meaningful for
aesthetic design.
Suggested/Internal Citation Key
Fontana:1999:AFF
#81:
D. J. Walton and D. S. Meek. Clothoidal splines, Computers & Graphics, 14 (1), pp. 95-100 (1990).
Keyword(s)
curves and surfaces
Suggested/Internal Citation Key
Walton:1990:CS
#82:
Anis Limaiem and François Trochu. Geometric algorithms for the intersection of curves and surfaces, Computers & Graphics, 19 (3), pp. 391-403 (May
1995). Pergamon Press / Elsevier Science. ISSN 0097-8493.
Suggested/Internal Citation Key
Limaiem:1995:GAF
#83:
Zhidong Guan and Jin Ling and Ning Tao and Xi Ping and Tang Rongxi. Study and application of physics-based deformable curves and surfaces, Computers &
Graphics, 21(3), pp. 305-313 (May 1997). Pergamon Press / Elsevier Science. ISSN 0097-8493.
Abstract
Physics-based deformable curve and surface modeling techniques are presented. The new approach is efficient and easy to use in many aspects of geometric design. Its
applications include the construction of N-sided patches, surface smooth joining, curve and surface fairing, etc.. The equations of motion for the deformable curves and
surfaces are derived using the Lagrangian mechanics and are solved by the finite element method. The constraints, such as the position of points on curves or surfaces, the
tangent plane and cross-derivative of surface are implemented through the penalty function methods.
Suggested/Internal Citation Key
Guan:1997:SAA
#84:
Frederic Cros and Philip J. Brock. A method for providing full interactive control of the shape of 3d curves and surfaces, Eurographics '88, pp. 443-455 (September
1988). North-Holland. Edited by D. A. Duce and P. Jancene.
Suggested/Internal Citation Key
Cros:1988:AMF
#85:
Christophe Rabut. Even Degree B-Spline Curves and Surfaces. A Note on the Paper "B-Spline Curves and Surfaces Viewed as Digital Filters" by A. Goshtasby, F. Cheng and
B. Barsky, CVGIP: Graphical Models and Image Processing, 54 (4), pp. 351-356 (July 1992).
Suggested/Internal Citation Key
Rabut:1992:EDB
#86:
L.-M. Reissell. Wavelet Multiresolution Representation of Curves and Surfaces, Graphical Models and Image Processing, 58 (3), pp. 198-217 (May 1996). Academic Press.
Suggested/Internal Citation Key
Reissell:1996:WMR
#87:
Gershon Elber and Myung-Soo Kim. Geometric Shape Recognition of Freeform Curves and Surfaces, Graphical Models and Image Processing, 59 (6), pp.
417-433 (November 1997). Academic Press.
Suggested/Internal Citation Key
Elber:1997:GSR
#88:
Jiwen Zhang. C-Bézier Curves and Surfaces, Graphical Models and Image Processing, 61(1), pp. 2-15 (January 1999). Academic Press.
Keyword(s)
C-Bézier curves, C-curves, C-B-splines, tensor product C-Bézier surfaces, Bézier curves, cubic curbes, B-splines, tensor product Bézier surfaces
Copyright
Copyright © 1999 Academic Press
Abstract
Using the same technique as for the C-B-splines, two other forms of C-Bézier curves and a reformed formula for the subdivisions are proposed. With these new forms,
C-Bézier curves can unify the processes for both the normal cases, and the limiting case (a -> 0) with precise results. Like the C-B-splines, a C-Bézier curve can be
approximated by its cubic Bézier curve in high accuracy. For any tensor product C-Bézier patch, a pair of its opposite sides could have different parameters of a. All this will
make the C-Bézier curves and surfaces more efficient in algorithms, more flexible in assembling and representing arcs, and will satisfy the demands of high precision in
engineering and fast calculation in computer display.
Suggested/Internal Citation Key
Zhang:1999:CCA
#89:
Thomas W. Sederberg and Jianmin Zheng and Kris Klimaszewski and Tor Dokken. Approximate Implicitization Using Monoid Curves and Surfaces, Graphical Models and
Image Processing, 61(4), pp. 177-198 (July 1999). Academic Press.
Copyright
Copyright © 1999 Academic Press
Abstract
This paper presents an approach to finding an approximate implicit equation and an approximate inversion map of a planar rational parametric curve or a rational parametric
surface. High accuracy of the approximation is achieved with a relatively small number of low-degree curve segments or surface patches. By using monoid curves and surfaces,
the method eliminates the undesirable singularities and "phantom" branches normally associated with implicit representation. The monoids are expressed in exact implicit and
parametric equations simultaneously, and upper bounds are derived for the approximate errors of implicitization and inversion equations.
Suggested/Internal Citation Key
Sederberg:1999:AIU
#90:
Martin Peternell. Geometric Properties of Bisector Surfaces, Graphical Models, 62(3), pp. 202-236 (May 2000). Academic Press. ISSN 1524-0703.
Copyright
Copyright © 2000 Academic Press
Abstract
This paper studies algebraic and geometric properties of curve-curve, curve-surface, and surface-surface bisectors. The computation is in general difficult since the bisector is
determined by solving a system of nonlinear equations. Geometric considerations will help us to determine several distinguished curve and surface pairs which possess
elementary computable bisectors. Emphasis is on low-degree rational curves and surfaces, since they are of particular interest in surface modeling.
Suggested/Internal Citation Key
Peternell:2000:GPO
#91:
Andrew Glassner. Planar Cubic Curves, Graphics Gems, pp. 575-578 (1990, Boston). Academic Press. Edited by Andrew S. Glassner. ISBN 0-12-286166-3.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Glassner:1990:PCC
#92:
Richard Rasala. Explicit Cubic Spline Interpolation Formulas, Graphics Gems, pp. 579-584 (1990, Boston). Academic Press. Edited by Andrew S. Glassner. ISBN
0-12-286166-3.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Rasala:1990:ECS
#93:
Julian Gomez. Fast Spline Drawing, Graphics Gems, pp. 585-586 (1990, Boston). Academic Press. Edited by Andrew S. Glassner. ISBN 0-12-286166-3.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Gomez:1990:FSD
#94:
Ronald Goldman. Some Properties of Bézier Curves, Graphics Gems, pp. 587-593 (1990, Boston). Academic Press. Edited by Andrew S. Glassner. ISBN 0-12-286166-3.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Goldman:1990:SPO
#95:
Bob Wallis. Tutorial on Forward Differencing, Graphics Gems, pp. 594-603 (1990, Boston). Academic Press. Edited by Andrew S. Glassner. ISBN 0-12-286166-3.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Wallis:1990:TOF
#96:
Ronald Goldman. Integration of Bernstein Basis Functions, Graphics Gems, pp. 604-606 (1990, Boston). Academic Press. Edited by Andrew S. Glassner. ISBN
0-12-286166-3.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Goldman:1990:IOB
#97:
Philip J. Schneider. Solving the Nearest-Point-on-Curve Problem, Graphics Gems, pp. 607-611, 787-796 (1990, Boston). Academic Press. Edited by Andrew S.
Glassner. ISBN 0-12-286166-3.
Keyword(s)
Curves and Surfaces
Abstract
includes code
Suggested/Internal Citation Key
Schneider:1990:STN
#98:
Philip J. Schneider. An Algorithm for Automatically Fitting Digitized Curves, Graphics Gems, pp. 612-626, 797-807 (1990, Boston). Academic Press. Edited by Andrew S.
Glassner. ISBN 0-12-286166-3.
Keyword(s)
Curves and Surfaces
Abstract
includes code
Suggested/Internal Citation Key
Schneider:1990:AAF
#99:
Doug Moore. Least-Squares Approximations To Bézier Curves and Surfaces, Graphics Gems II, pp. 406-411 (1991, Boston). Academic Press. Edited by James Arvo. ISBN
0-12-064481-9.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Moore:1991:LAT
#100:
Ken Shoemake. Beyond Bézier Curves, Graphics Gems II, pp. 412-416 (1991, Boston). Academic Press. Edited by James Arvo. ISBN 0-12-064481-9.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Shoemake:1991:BBC
#101:
John Schlag. A Simple Formulation for Curve Interpolation with Variable Control Point Approximation, Graphics Gems II, pp. 417-419 (1991, Boston). Academic
Press. Edited by James Arvo. ISBN 0-12-064481-9.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Schlag:1991:ASF
#102:
Terence Lindgren. Symmetric Evaluation of Polynomials, Graphics Gems II, pp. 420-423 (1991, Boston). Academic Press. Edited by James Arvo. ISBN 0-12-064481-9.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Lindgren:1991:SEO
#103:
Hans-Peter Seidel. Menelaus's Theorem, Graphics Gems II, pp. 424-427 (1991, Boston). Academic Press. Edited by James Arvo. ISBN 0-12-064481-9.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Seidel:1991:MT
#104:
Hans-Peter Seidel. Geometrically Continuous Cubic Bézier Curves, Graphics Gems II, pp. 428-434 (1991, Boston). Academic Press. Edited by James Arvo. ISBN
0-12-064481-9.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Seidel:1991:GCC
#105:
Christopher J. Musial. A Good Straight-Line Approximation of a Circular Arc, Graphics Gems II, pp. 435-439, 617 (1991, Boston). Academic Press. Edited by James
Arvo. ISBN 0-12-064481-9.
Keyword(s)
Curves and Surfaces
Abstract
includes code
Suggested/Internal Citation Key
Musial:1991:AGS
#106:
Alan W. Paeth. Great Circle Plotting, Graphics Gems II, pp. 440-445 (1991, Boston). Academic Press. Edited by James Arvo. ISBN 0-12-064481-9.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Paeth:1991:GCP
#107:
Xiaolin Wu. Fast Anti-Aliased Circle Generation, Graphics Gems II, pp. 446-450 (1991, Boston). Academic Press. Edited by James Arvo. ISBN 0-12-064481-9.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Wu:1991:FAC
#108:
Paul H. C. Eilers. Smoothing and Interpolation with Finite Differences, Graphics Gems IV, pp. 241-250 (1994, Boston). Academic Press. Edited by Paul S. Heckbert. ISBN
0-12-336155-9.
Keyword(s)
Curves and Surfaces
Abstract
includes code
Suggested/Internal Citation Key
Eilers:1994:SAI
#109:
Phillip Barry. Knot Insertion using Forward Differences, Graphics Gems IV, pp. 251-255 (1994, Boston). Academic Press. Edited by Paul S. Heckbert. ISBN
0-12-336155-9.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Barry:1994:KIU
#110:
Chandrajit Bajaj. Converting a Rational Curve to a Standard Rational Bernstein-Bézier Representation, Graphics Gems IV, pp. 256-260 (1994, Boston). Academic
Press. Edited by Paul S. Heckbert. ISBN 0-12-336155-9.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Bajaj:1994:CAR
#111:
R. Victor Klassen. Intersecting Parametric Cubic Curves by Midpoint Subdivision, Graphics Gems IV, pp. 261-277 (1994, Boston). Academic Press. Edited by Paul S.
Heckbert. ISBN 0-12-336155-9.
Keyword(s)
Curves and Surfaces
Abstract
includes code
Suggested/Internal Citation Key
Klassen:1994:IPC
#112:
Dani Lischinski. Converting Rectangular Patches into Bézier Triangles, Graphics Gems IV, pp. 278-285 (1994, Boston). Academic Press. Edited by Paul S. Heckbert. ISBN
0-12-336155-9.
Keyword(s)
Curves and Surfaces
Abstract
includes code
Suggested/Internal Citation Key
Lischinski:1994:CRP
#113:
John W. Peterson. Tessellation of NURB Surfaces, Graphics Gems IV, pp. 286-320 (1994, Boston). Academic Press. Edited by Paul S. Heckbert. ISBN 0-12-336155-9.
Keyword(s)
Curves and Surfaces
Abstract
includes code
Suggested/Internal Citation Key
Peterson:1994:TON
#114:
Ching-Kuang Shene. Equations of Cylinders and Cones, Graphics Gems IV, pp. 321-323 (1994, Boston). Academic Press. Edited by Paul S. Heckbert. ISBN
0-12-336155-9.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Shene:1994:EOC
#115:
Jules Bloomenthal. An Implicit Surface Polygonizer, Graphics Gems IV, pp. 324-349 (1994, Boston). Academic Press. Edited by Paul S. Heckbert. ISBN 0-12-336155-9.
Keyword(s)
Curves and Surfaces
Abstract
includes code
Suggested/Internal Citation Key
Bloomenthal:1994:AIS
#116:
Ronald Goldman. Identities for the Univariate, Bivariate Bernstein Basis Fcns, Graphics Gems V, pp. 149-162 (1995, Boston). Academic Press. Edited by Alan W.
Paeth. ISBN 0-12-543455-3.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Goldman:1995:IFT
#117:
Ronald Goldman. Identities for the B-Spline Basis Functions, Graphics Gems V, pp. 163-167 (1995, Boston). Academic Press. Edited by Alan W. Paeth. ISBN
0-12-543455-3.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Goldman:1995:IFTB
#118:
Ken Turkowski. Circular Arc Subdivision, Graphics Gems V, pp. 168-172 (1995, Boston). Academic Press. Edited by Alan W. Paeth. ISBN 0-12-543455-3.
Keyword(s)
Curves and Surfaces
Abstract
includes code
Suggested/Internal Citation Key
Turkowski:1995:CAS
#119:
Luiz Henrique de Figueiredo. Adaptive Sampling of Parametric Curves, Graphics Gems V, pp. 173-178 (1995, Boston). Academic Press. Edited by Alan W. Paeth. ISBN
0-12-543455-3.
Keyword(s)
Curves and Surfaces
Abstract
includes code
Suggested/Internal Citation Key
deFigueiredo:1995:ASO
#120:
Jaewoo Ahn. Fast Generation of Ellipsoids, Graphics Gems V, pp. 179-190 (1995, Boston). Academic Press. Edited by Alan W. Paeth. ISBN 0-12-543455-3.
Keyword(s)
Curves and Surfaces
Abstract
includes code
Suggested/Internal Citation Key
Ahn:1995:FGO
#121:
Chandrajit Bajaj. Sparse Smooth Connection Between Bézier/B-Spline Curves, Graphics Gems V, pp. 191-198 (1995, Boston). Academic Press. Edited by Alan W.
Paeth. ISBN 0-12-543455-3.
Keyword(s)
Curves and Surfaces
Suggested/Internal Citation Key
Bajaj:1995:SSC
#122:
Jens Gravesen. The Length of Bézier Curves, Graphics Gems V, pp. 199-205 (1995, Boston). Academic Press. Edited by Alan W. Paeth. ISBN 0-12-543455-3.
Keyword(s)
Curves and Surfaces
Abstract
includes code
Suggested/Internal Citation Key
Gravesen:1995:TLO
#123:
Robert D. Miller. Quick and Simple Bézier Curve Drawing, Graphics Gems V, pp. 206-209 (1995, Boston). Academic Press. Edited by Alan W. Paeth. ISBN 0-12-543455-3.
Keyword(s)
Curves and Surfaces
Abstract
includes code
Suggested/Internal Citation Key
Miller:1995:QAS
#124:
Ken Shoemake. Linear Form Curves, Graphics Gems V, pp. 210-223 (1995, Boston). Academic Press. Edited by Alan W. Paeth. ISBN 0-12-543455-3.
Keyword(s)
Curves and Surfaces
Abstract
includes code
Suggested/Internal Citation Key
Shoemake:1995:LFC
#125:
Brian A. Barsky and Alain Fournier. Computational techniques for parametric curves and surfaces, Graphics Interface '82, pp. 57-71 (May 1982).
Suggested/Internal Citation Key
Barsky:1982:CTF
#126:
Chandrajit L. Bajaj and Andrew V. Royappa. Parameterization in finite precision, Graphics Interface '92, pp. 29-36 (May 1992). Canadian Information Processing Society.
Keyword(s)
curves and surfaces, geometric modeling, numerical methods, computational algebraic geometry
Suggested/Internal Citation Key
Bajaj:1992:PIF
#127:
Wayne Liu and Stephen Mann. Programming Support for Blossoming, Graphics Interface '96, pp. 95-106 (May 1996). Canadian Human-Computer Communications
Society. Edited by Wayne A. Davis and Richard Bartels. ISBN 0-9695338-5-3.
URL
This article is available to download (usually PDF or PostScript).
It may be freely available, or require membership in an organization's digital library.
Keyword(s)
blossoms, curves and surfaces, graphics data structures and data types, software
Abstract
A C++ library has been created to facilitate prototyping of curve and surface modeling techniques. The library provides blossoming datatypes to support creation of modeling
techniques based on blossoming analysis. The datatypes have efficient operations that are generalizations of important CAGD algorithms and can be used to implement many
algorithms. Most importantly, the library is able to interoperate with user-supplied datatypes or routines to create complex modeling techniques.
Suggested/Internal Citation Key
Liu:1996:PSF
#128:
Brian A. Barsky and Tony D. DeRose. The Beta2-spline: A Special Case of the Beta-spline Curve and Surface Representation, IEEE Computer Graphics &
Applications, 5(9), pp. 46-58 (September 1985).
Keyword(s)
splines, curves and surfaces, design and modeling, graphics, differential geometry, algorithms, CAD/CAM, subdivision
Abstract
Correction published in Letter to the Editor, IEEE Computer Graphics and Applications, Vol. 7, No. 3, March 1987, p. 15. Earlier version of article published as Tech. Report
No. UCB/CSD 83/152, Computer Science Division, Electrical Engineering and Computer Sciences Department, University of California, Berkeley, California, USA.
(November, 1983).
Suggested/Internal Citation Key
Barsky:1985:TBA
#129:
Gerald Farin and Nickolas Sapidis. Curvature and the Fairness of Curves and Surfaces, IEEE Computer Graphics & Applications, 9 (2), pp. 52-57 (March 1989).
Keyword(s)
fairing curves
Suggested/Internal Citation Key
Farin:1989:CAT
#130:
Les Piegl. On NURBS: A Survey, IEEE Computer Graphics & Applications, 11(1), pp. 55-71 (January 1991).
Keyword(s)
nurbs, curves and surfaces
Abstract
Survey on nurbs, definitions, properties and control.
Suggested/Internal Citation Key
Piegl:1991:ONA
#131:
Jonathan Yen and Susan Spach and Mark Smith and Ron Pulleyblank. Parallel Boxing in B-Spline Intersection, IEEE Computer Graphics & Applications, 11(1), pp.
72-79 (January 1991).
Keyword(s)
curves and surfaces, boxing
Abstract
Use oriented boxes around B-spline to speed up intersection in a parallel algorithm.
Suggested/Internal Citation Key
Yen:1991:PBI
#132:
Christoph M. Hoffmann. Implicit curves and surfaces in CAGD, IEEE Computer Graphics & Applications, 13(1), pp. 79-88 (January 1993).
Abstract
Conversion between parametric and implicit forms has always been possible, but practical problems have forced researchers to explore alternatives such as deferring or
sidestepping symbolic computation.
Suggested/Internal Citation Key
Hoffmann:1993:ICA
#133:
Hans-Peter Seidel. An introduction to polar forms, IEEE Computer Graphics & Applications, 13(1), pp. 38-46 (January 1993).
Abstract
Polar forms simplify the construction of polynomial and piecewise-polynomial curves and surfaces and lead to new surface representations and algorithms.
Suggested/Internal Citation Key
Seidel:1993:AIT
#134:
Ron Goldman. The Ambient Spaces of Computer Graphics and Geometric Modeling, IEEE Computer Graphics & Applications, 20(2), pp. 76-84 (March/April 2000). ISSN
0272-1716.
Copyright
Copyright © 2000 IEEE
Abstract
Four types of ambient mathematical spaces underlie the algebra and geometry of computer graphics and geometric modeling: vector spaces, affine spaces, projective spaces, and
Grassmann spaces. This article clarifies the relationships between these different ambient spaces and explains as well how they support the construction of the standard
polynomial and rational freeform curves and surfaces of geometric design.
Suggested/Internal Citation Key
Goldman:2000:TAS
#135:
Les A. Piegl and Wayne Tiller. Reducing Control Points in Surface Interpolation, IEEE Computer Graphics & Applications, 20(5), pp. 70-74 (September - October
2000). ISSN 0272-1716.
URL
This article is available to download (usually PDF or PostScript).
It may be freely available, or require membership in an organization's digital library.
Keyword(s)
Data interpolation, skinning, B-splines, curves and surfaces, algorithms
Copyright
Copyright © 2000 IEEE
Abstract
A method for interpolating rows of data points with B-spline surfaces is presented. In each row the number points can differ, requiring a skinning-type operator to pass a surface
through the points. To avoid data explosion as a result of knot merging, we introduce a new curve interpolation method that uses knots from a given input knot vector.
Depending on the initial knot vector and how it is updated during interpolation from row to row, the new method reduces the number of surface control points by 60-97%.
Suggested/Internal Citation Key
Piegl:2000:RCP
#136:
Andrew J. Hanson. Constrained Optimal Framings of Curves and Surfaces Using Quaternion Gauss Maps, IEEE Visualization '98, pp. 375-382 (October 1998). IEEE. Edited
by David Ebert and Hans Hagen and Holly Rushmeier. ISBN 0-8186-9176-X.
Suggested/Internal Citation Key
Hanson:1998:COF
#137:
M. Pratt. Parametric curves and surfaces as used in computer aided design, The Mathematics of Surfaces, pp. 19-46 (1986). Clarendon Press. Edited by J. Gregory.
Suggested/Internal Citation Key
Pratt:1986:PCA
#138:
A. Ball. The parametric representation of curves and surfaces using rational polynomial functions, The Mathematics of Surfaces II, pp. 39-62 (1987). Oxford University
Press. Edited by R. Martin.
Suggested/Internal Citation Key
Ball:1987:TPR
#139:
M. Sabin. Envelope curves and surfaces, The Mathematics of Surfaces II, pp. 413-418 (1987). Oxford University Press. Edited by R. Martin.
Suggested/Internal Citation Key
Sabin:1987:ECA
#140:
B. Dahlberg and B. Johansson. Envelope curves and surfaces, The Mathematics of Surfaces II, pp. 419-426 (1987). Oxford University Press. Edited by R. Martin.
Suggested/Internal Citation Key
Dahlberg:1987:ECA
#141:
T. Sederberg. Implicit and parametric curves and surfaces for computer aided geometric design, (1983). Mech. Eng., Purdue U..
Suggested/Internal Citation Key
Sederberg:1983:IAP
#142:
D. Kim. Cones on Bézier curves and surfaces, (1990). Industrial and Operations Engineering Dept, U. of Michigan at Ann Arbor.
Suggested/Internal Citation Key
Kim:1990:COB
#143:
Leon A. Shirman. Construction of smooth curves and surfaces from polyhedral models, pp. 195 (1990). Computer Science Division (EECS), University of California,
Berkeley.
Keyword(s)
polyhedron
Abstract
Also as tech report no UCB/CSD 90/602.
Suggested/Internal Citation Key
Shirman:1990:COS
#144:
Brian A. Barsky and Tony D. DeRose and Mark D. Dippe. An Adaptive Subdivision Method with Crack Prevention for Rendering Beta-spline Objects, (March
1987). Computer Science Division, Electrical Engineering and Computer Sciences Department, University of California, Berkeley, California, USA..
Keyword(s)
splines, curves and surfaces, design and modeling, graphics, differential geometry, algorithms, CAD/CAM, subdivision
Suggested/Internal Citation Key
Barsky:1987:AAS
#145:
Brian A. Barsky. A Study of the Parametric Uniform B-spline Curve and Surface Representations, (May 1983). Computer Science Division, Electrical Engineering and
Computer Sciences Department, University of California, Berkeley, California, USA..
Keyword(s)
splines, curves and surfaces, design and modeling, algorithms, CAD/CAM
Suggested/Internal Citation Key
Barsky:1983:ASO
#146:
Richard F. Riesenfeld and Elaine Cohen and Russell D. Fish and Spencer W. Thomas and Elizabeth S. Cobb and Brian A. Barsky and Dino L. Schweitzer and Jeffrey M.
Lane. Using the Oslo Algorithm as a Basis for CAD/CAM Geometric Modelling, Proceedings of the Second Annual NCGA National Conference, pp. 345-356 (June
1981). National Computer Graphics Association, Inc..
Keyword(s)
splines, curves and surfaces, design and modeling, graphics, algorithms, CAD/CAM
Suggested/Internal Citation Key
Riesenfeld:1981:UTO
#147:
Brian A. Barsky. End Conditions and Boundary Conditions for Uniform B-spline Curve and Surface Representations, Computers in Industry, 3 (1, 2), pp. 17-29 (March
1982).
Keyword(s)
splines, curves and surfaces, design and modeling, CAD/CAM
Suggested/Internal Citation Key
Barsky:1982:ECA
#148:
S. Ocken and Jacob T. Schwartz and M. Sharir. Precise Implementation of CAD Primitives Using Rational Parameterizations of Standard Surfaces, Solid Modeling by
Computers, pp. 259-273 (1983). Plenum Press. Edited by Mary S. Pickett and John W. Boyse.
Keyword(s)
splines, curves and surfaces, design and modeling, differential geometry, algorithms, CAD/CAM
Suggested/Internal Citation Key
Ocken:1983:PIO
#149:
T. Sederberg and D. Anderson and R. Goldman. Implicit representation of parametric curves and surfaces, Computer Vision, Graphics, and Image Processing, 28 (1),
pp. 72-84 (1984).
Suggested/Internal Citation Key
Sederberg:1984:IRO
#150:
G. Cisneros and N. Garcia. Three-Dimensional Pictures with Curves and Surfaces in Parametric Coordinates, Proceedings MELECON '85, Mediterranean Electrotechnical
Conference (4 vols), 2 (), pp. 213-216 (1985). IEEE. Edited by A. Luque and A. R. Figueiras Vidal and J. M. R. Delgado.
Keyword(s)
image representations
Suggested/Internal Citation Key
Cisneros:1985:TPW
#151:
Tim N. T. Goodman and Keith Unsworth. Generation of Beta-spline Curves Using a Recurrence Relation, Fundamental Algorithms for Computer Graphics, pp.
325-357 (1985). Springer-Verlag. Edited by Rae A. Earnshaw.
Keyword(s)
curves and surfaces
Suggested/Internal Citation Key
Goodman:1985:GOB
#152:
J. Hoschek. Dual Bézier Curves and Surfaces, Surfaces in Computer Aided Geometric Design, pp. 147-156 (1985). North-Holland. Edited by R. Barnhill and W. Boehm.
Suggested/Internal Citation Key
Hoschek:1985:DBC
#153:
Klaus Hollig. Geometric Continuity of Spline Curves and Surfaces, (June 1986, Madison, WI). Computer Sciences Department, University of Wisconsin.
Abstract
Abstract: We review beta-spline theory for curves and show how some of the concepts can be extended to surfaces. Our approach is based on the Bézier form for piecewise
polynomials which yields simple geometric characterizations of smoothness constraints and shape parameters. For curves most of the standard "spline calculus" has been
developed. We discuss in particular the construction of B-splines, the conversion for B-spline to Bézier representation and interpolation algorithms. A comparable theory for
spline surfaces for general meshes does at present not exist. We merely describe how to join triangular and rectangular patches and discuss the corresponding beta-spline
constraints in terms of the Bézier representation.
Suggested/Internal Citation Key
Hollig:1986:GCO
#154:
W. Boehm. Smooth curves and surfaces, Geometric Modeling: Algorithms and New Trends, pp. 175-184 (1987). SIAM, Philadelphia. Edited by G. Farin.
Suggested/Internal Citation Key
Boehm:1987:SCA
#155:
A. Jones. Shape control of curves and surfaces through constrained optimization, Geometric Modeling: Algorithms and New Trends, pp. 265-279 (1987). SIAM,
Philadelphia. Edited by G. Farin.
Suggested/Internal Citation Key
Jones:1987:SCO
#156:
A. Schwartz. Subdividing Bézier curves and surfaces, Geometric Modeling: Algorithms and New Trends, pp. 55-66 (1987). SIAM, Philadelphia. Edited by G. Farin.
Suggested/Internal Citation Key
Schwartz:1987:SB
#157:
W. Boehm. Differential Geometry I, (1988).
Abstract
Chapter 11 in G. Farin: Curves and Surfaces for Computer Aided Geometric Design, Academic Press
Suggested/Internal Citation Key
Boehm:1988:DGI
#158:
F. Yamaguchi. Curves and Surfaces in Computer Aided Geometric Design, (1988). Springer.
Suggested/Internal Citation Key
Yamaguchi:1988:CAS
#159:
A. Cavaretta and C. Micchelli. The design of curves and surfaces by subdivision algorithms, Mathematical Methods in Computer Aided Geometric Design, pp.
115-154 (1989). Academic Press. Edited by T. Lyche and L. Schumaker.
Suggested/Internal Citation Key
Cavaretta:1989:TDO
#160:
R. Goldman and B. Barsky. On beta-continuous functions and their application to the construction of geometrically continuous curves and surfaces, Mathematical Methods
in Computer Aided Geometric Design, pp. 299-312 (1989). Academic Press. Edited by T. Lyche and L. Schumaker.
Suggested/Internal Citation Key
Goldman:1989:OBF
#161:
C. M. Hoffmann. Geometric and Solid Modeling: An Introduction, (1989). Morgan Kaufmann. ISBN: 1-55860-067-1 .
Keyword(s)
cad
Abstract
Probably the best book on the subject written so far if you want to get into the details of the subject. As well as showing how to write robust boundary representation modeller
code, it is strong on the algebraic approach to curves and surfaces.
Suggested/Internal Citation Key
Hoffmann:1989:GAS
#162:
Gerald Farin. Curves and Surfaces for Computer Aided Geometric Design, pp. 464 (1990). Academic Press.
Abstract
2nd edition, ISBN: 0-12-249051
Suggested/Internal Citation Key
Farin:1990:CAS
#163:
Natasha Oza. An interactive tool to illustrate the basic principles of Bézier and B-spline curves and surfaces, (August 1990). Rensselaer Polytechnic Institute.
Abstract
Advisor: M. Wozny
Suggested/Internal Citation Key
Oza:1990:AIT
#164:
T. DeRose. Rational Bézier curves and surfaces on projective domains, NURBS for Curve and Surface Design, pp. 35-46 (1991). SIAM. Edited by G. Farin.
Suggested/Internal Citation Key
DeRose:1991:RBC
#165:
X. S. Gao and S. C. Chou. On the normal parametrization of curves and surfaces, Internat. J. Comput. Geom. Appl., 1 (2), pp. 125-136 (1991).
Keyword(s)
normal parametric equation, inversion map, conic, conicoid, computer modeling
Suggested/Internal Citation Key
Gao:1991:OTN
#166:
E. Brechner. General offset curves and surfaces, Geometry Processing for Design and Manufacturing, pp. 101-121 (1992). SIAM, Philadelphia. Edited by R. E. Barnhill.
Suggested/Internal Citation Key
Brechner:1992:GOC
#167:
Lyle H. Ramshaw. Bézier and B-splines as multiaffine maps, Theoretical Foundations of Computer Graphics and CAD, NATO ASI, F40 (), pp.
757-776 (1988). Springer-Verlag. Edited by R. A. Earnshaw.
Keyword(s)
curves and surfaces, bezier triangle, computer aided geometric design, de boor algorithm, de casteljau algorithm, interpolation, multilinearity
Suggested/Internal Citation Key
Ramshaw:1988:BAB
#168:
Atsushi Yamada and Tomotake Furuhata and Kenji Shimada and Ko-Hsiu Hou. A Discrete Spring Model for Genarating Fair Curves and Surfaces, Pacific Graphics
'99, (October 1999, Seoul, Korea).
Copyright
Copyright © 1999 IEEE
Suggested/Internal Citation Key
Yamada:1999:ADS
#169:
J. A. Brewer and D. C. Anderson. Visual Interaction with Overhauser Curves and Surfaces, Computer Graphics (Proceedings of SIGGRAPH 77), 11 (2), pp. 132-137 (July
1977, San Jose, California). Edited by James George.
Keyword(s)
cardinal spline
Copyright
Copyright © 1977 Association for Computing Machinery
Suggested/Internal Citation Key
Brewer:1977:VIW
#170:
J. N. England. A system for interactive modeling of physical curved surface objects, Computer Graphics (Proceedings of SIGGRAPH 78), 12 (3), pp. 336-340 (August
1978, Atlanta, Georgia).
Keyword(s)
object modelling, curves and surfaces, design and modeling, graphics, systems applications
Copyright
Copyright © 1978 Association for Computing Machinery
Suggested/Internal Citation Key
England:1978:ASF
#171:
Loren C. Carpenter. Computer Rendering of Fractal Curves and Surfaces, Computer Graphics (Proceedings of SIGGRAPH 80), 14 (3), pp. 109 (July 1980, Seattle,
Washington).
Copyright
Copyright © 1980 Association for Computing Machinery
Suggested/Internal Citation Key
Carpenter:1980:CRO
#172:
Sheue-Ling Lien and Michael Shantz and Vaughan Pratt. Adaptive Forward Differencing for Rendering Curves and Surfaces, Computer Graphics (Proceedings of SIGGRAPH
87), 21 (4), pp. 111-118 (July 1987, Anaheim, California). Edited by Maureen C. Stone.
Keyword(s)
image synthesis, adaptive forward differencing, parametric curves and surfaces
Copyright
Copyright © 1987 Association for Computing Machinery
Suggested/Internal Citation Key
Lien:1987:AFD
#173:
Barry Joe. Discrete Beta-Splines, Computer Graphics (Proceedings of SIGGRAPH 87), 21 (4), pp. 137-144 (July 1987, Anaheim, California). Edited by Maureen C. Stone.
Keyword(s)
b-splines, subdivision, knot refinement, geometric continuity, computer-aided geometric design, curves and surfaces, design and modeling, CAD/CAM
Copyright
Copyright © 1987 Association for Computing Machinery
Suggested/Internal Citation Key
Joe:1987:DB
#174:
Sheue-Ling Chang and Michael Shantz and Robert Rocchetti. Rendering Cubic Curves and Surfaces with Integer Adaptive Forward Differencing, Computer Graphics
(Proceedings of SIGGRAPH 89), 23 (3), pp. 157-166 (July 1989, Boston, Massachusetts). Edited by Jeffrey Lane.
Keyword(s)
adaptive forward differencing, parametric curve, texture
Copyright
Copyright © 1989 Association for Computing Machinery
Suggested/Internal Citation Key
Chang:1989:RCC
#175:
Thomas W. Sederberg and Falai Chen. Implicitization Using Moving Curves and Surfaces, Proceedings of SIGGRAPH 95, Computer Graphics Proceedings, Annual
Conference Series, pp. 301-308 (August 1995, Los Angeles, California). Addison Wesley. Edited by Robert Cook. ISBN 0-201-84776-0.
URL
This article is available to download (usually PDF or PostScript).
It may be freely available, or require membership in an organization's digital library.
Keyword(s)
Bézier patches, implicitization, base points
Copyright
Copyright © 1995 Association for Computing Machinery
Abstract
This paper presents a radically new approach to the century old problem of computing the implicit equation of a parametric surface. For surfaces without base points, the new
method expresses the implicit equation in a determinant which is one fourth the size of the conventional expression based on Dixon's resultant. If base points do exist, previous
implicitization methods either fail or become much more complicated, while the new method actually simplifies. The new method is illustrated using the bicubic patches from
Newell's teapot model. Dixon's method can successfully implicitize only 8 of those 32 patches, ex-pressing the implicit equation as an 18 ? 18 determinant. The new method
successfully implicitizes all 32 of the patches. Four of the implicit equations can be written as 3 ? 3 determinants, eight can be written as 4 ? 4 determinants, and the remaining 20
implicit equations can be written using 9 ? 9 determinants.
Suggested/Internal Citation Key
Sederberg:1995:IUM
#176:
Carole Blanc and Christophe Schlick. X-Splines: A Spline Model Designed for the End-User, Proceedings of SIGGRAPH 95, Computer Graphics Proceedings, Annual
Conference Series, pp. 377-386 (August 1995, Los Angeles, California). Addison Wesley. Edited by Robert Cook. ISBN 0-201-84776-0.
URL
This article is available to download (usually PDF or PostScript).
It may be freely available, or require membership in an organization's digital library.
Copyright
Copyright © 1995 Association for Computing Machinery
Abstract
This paper presents a new model of spline curves and surfaces. The main characteristic of this model is that it has been created from scratch by using a kind of mathematical
engineering process. In a first step, a list of specifications was established. This list groups all the properties that a spline model should contain in order to appear intuitive to a
non-mathematician end-user. In a second step, a new family of blending functions was derived, trying to fulfill as many items as possible of the previous list. Finally, the
degrees of freedom offered by the model have been reduced to provide only shape parameters that have a visual interpretation on the screen. The resulting model includes many
classical properties such as affine and perspective invariance, convex hull, variation diminution, local control and C2 / G2 or C2 / G0 continuity. But it also includes
original features such as a continuum between B-splines and Catmull-Rom splines, or the ability to define approximation zones and interpolation zones in the same curve or
surface.
Suggested/Internal Citation Key
Blanc:1995:XAS
#177:
Ravi Ramamoorthi and James Arvo. Creating Generative Models From Range Images, Proceedings of SIGGRAPH 99, Computer Graphics Proceedings, Annual Conference
Series, pp. 195-204 (August 1999, Los Angeles, California). Addison Wesley Longman. Edited by Alyn Rockwood. ISBN 0-20148-560-5.
URL
This article is available to download (usually PDF or PostScript).
It may be freely available, or require membership in an organization's digital library.
Keyword(s)
Generative Models, Range Images, Curves and Surfaces, Procedural Modeling
Copyright
Copyright © 1999 Association for Computing Machinery
Abstract
We describe a new approach for creating concise high-level generative models from range images or other approximate representations of real objects. Using data from a variety
of acquisition techniques and a user-defined class of models, our method produces a compact object representation that is intuitive and easy to edit. The algorithm has two
inter-related phases: recognition, which chooses an appropriate model within a user-specified hierarchy, and parameter estimation, which adjusts the model to best fit the data.
Since the approach is model-based, it is relatively insensitive to noise and missing data. We describe practical heuristics for automatically making tradeoffs between simplicity
and accuracy to select the best model in a given hierarchy. We also describe a general and efficient technique for optimizing a model by refining its constituent curves. We
demonstrate our approach for model recovery using both real and synthetic data and several generative model hierarchies.
Suggested/Internal Citation Key
Ramamoorthi:1999:CGM
#178:
Shreeram S. Abhyankar. Parametrization of curves and surfaces, Curves and Surfaces in Computer Vision and Graphics (Proceedings of SPIE), 1251 (), pp.
122-129 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Abhyankar:1990:POC
#179:
D. J. Amalraj and K. Eswaran and N. Sundararajan. Determination of the curvature of surfaces and surface profiles, Curves and Surfaces in Computer Vision and Graphics
(Proceedings of SPIE), 1251 (), pp. 369-379 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Amalraj:1990:DOT
#180:
Chanderjit Bajaj. G1 interpolation using piecewise quadric and cubic surfaces, Curves and Surfaces in Computer Vision and Graphics (Proceedings of SPIE), 1251 (), pp.
82-93 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Keyword(s)
g1
Suggested/Internal Citation Key
Bajaj:1990:IU
#181:
Pierre E. Bézier. CAD/CAM in the French automobile industry, Curves and Surfaces in Computer Vision and Graphics (Proceedings of SPIE), 1251 (), pp.
2-9 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Bezier:1990:CIT
#182:
J. M. Blackledge. A method of incorporating prior information on the structure of random fractal surfaces, Curves and Surfaces in Computer Vision and Graphics (Proceedings
of SPIE), 1251 (), pp. 293 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Blackledge:1990:AMO
#183:
Li-Dong Cai. Approximating a surface up to curvature signs using the depth data alone, Curves and Surfaces in Computer Vision and Graphics (Proceedings of SPIE), 1251
(), pp. 254-260 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Cai:1990:AAS
#184:
Po-Rong Chang and Share-Young Lee. Partitioning and mapping B-spline surface fitting algorithm into fixed size VLSI arrays, Curves and Surfaces in Computer Vision and
Graphics (Proceedings of SPIE), 1251 (), pp. 106-110 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Chang:1990:PAM
#185:
Leonard A. Ferrari and Martine J. Silbermann and P. V. Sankar. Efficient curve and surface generation using high order differencing, Curves and Surfaces in Computer Vision
and Graphics (Proceedings of SPIE), 1251 (), pp. 262-271 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Ferrari:1990:ECA
#186:
Rui J. P. de Figueiredo and Nasser Kehtarnavaz. Blending functions for interpolation of networks of curves, Curves and Surfaces in Computer Vision and Graphics
(Proceedings of SPIE), 1251 (), pp. 41-43 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Keyword(s)
lagrange blending function, cardinal spline blending function, sinc blending function
Suggested/Internal Citation Key
Figueiredo:1990:BFF
#187:
John C. Hart and Alan Norton. Use of curves in rendering fractures, Curves and Surfaces in Computer Vision and Graphics (Proceedings of SPIE), 1251 (), pp.
322-328 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Hart:1990:UOC
#188:
Yuh-Tay Liow. A contour tracing algorithm that preserves common boundaries between regions, Curves and Surfaces in Computer Vision and Graphics (Proceedings of
SPIE), 1251 (), pp. 208-218 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Liow:1990:ACT
#189:
Michael Lounsbery and Charles Loop and Stephen Mann and David Meyers and James Painter and Tony DeRose and Kenneth Sloan. Testbed for the comparison of parametric
surface methods, Curves and Surfaces in Computer Vision and Graphics (Proceedings of SPIE), 1251 (), pp. 94-105 (1990). Edited by L. A. Ferrari and R. J. P. de
Figueiredo.
Suggested/Internal Citation Key
Lounsbery:1990:TFT
#190:
Akbary-Safa Mahnaz and Ibrahim I. Esat and Colin B. Besant. An efficient algorithm for error elimination from surface measurement generated by a mechanical probe, Curves
and Surfaces in Computer Vision and Graphics (Proceedings of SPIE), 1251 (), pp. 305-313 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Mahnaz:1990:AEA
#191:
Dinesh Manocha and John F. Canny. Polynomial parametrizations for rational curves, Curves and Surfaces in Computer Vision and Graphics (Proceedings of SPIE), 1251 (),
pp. 151-162 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Manocha:1990:PPF
#192:
Vishal Markandey and R. J. P. de Figueiredo. Graph-algebraic approach to 3D object representation, Curves and Surfaces in Computer Vision and Graphics (Proceedings of
SPIE), 1251 (), pp. 346-356 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Markandey:1990:GAT
#193:
A. Le Mehaute and Florencio I. Utreras. Shape preserving interpolating subdivision, Curves and Surfaces in Computer Vision and Graphics (Proceedings of SPIE), 1251 (),
pp. 74-81 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Mehaute:1990:SPI
#194:
S. P. Mudur and D. R. Khandekar. An interactive system for quick modeling of aircraft surfaces, Curves and Surfaces in Computer Vision and Graphics (Proceedings of
SPIE), 1251 (), pp. 314-320 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Mudur:1990:AIS
#195:
Hans-Peter Seidel. Symmetric algorithms for curves and surfaces, Curves and Surfaces in Computer Vision and Graphics (Proceedings of SPIE), 1251 (), pp.
18-29 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Keyword(s)
bpatch, bspline
Suggested/Internal Citation Key
Seidel:1990:SAF
#196:
G. R. Shevare and S. P. Mudur. Constrained-interior interpolating surfaces, Curves and Surfaces in Computer Vision and Graphics (Proceedings of SPIE), 1251 (), pp.
111-120 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Shevare:1990:CIS
#197:
M. J. Silbermann. High speed implementation of nonuniform rational B-splines (NURBS), Curves and Surfaces in Computer Vision and Graphics (Proceedings of
SPIE), 1251 (), pp. 338-345 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Silbermann:1990:HSI
#198:
Sunny Sunthankar. A solid object digitizing system, Curves and Surfaces in Computer Vision and Graphics (Proceedings of SPIE), 1251 (), pp. 329-337 (1990). Edited by
L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Sunthankar:1990:ASO
#199:
Boaz J. Super and Alan C. Bovik. Optimally localized estimation of the fractal dimension, Curves and Surfaces in Computer Vision and Graphics (Proceedings of SPIE), 1251
(), pp. 357-368 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Super:1990:OLE
#200:
Leonardo Traversoni. Delaunay's tetrahedronalization: An efficient algorithm for 3D triangulation, Curves and Surfaces in Computer Vision and Graphics (Proceedings of
SPIE), 1251 (), pp. 56-61 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Keyword(s)
convex polygons
Suggested/Internal Citation Key
Traversoni:1990:DTA
#201:
Joe Warren and Suresh Lodha. Free-form quadric surface patches, Curves and Surfaces in Computer Vision and Graphics (Proceedings of SPIE), 1251 (), pp.
30-40 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Warren:1990:FQS
#202:
Chris K. Wu and Peter L. Weiland and John B. Cheatham. A computer graphics testbed for developing and testing laser imaging algorithms, Curves and Surfaces in Computer
Vision and Graphics (Proceedings of SPIE), 1251 (), pp. 294-304 (1990). Edited by L. A. Ferrari and R. J. P. de Figueiredo.
Suggested/Internal Citation Key
Wu:1990:ACG
#203:
Laurent Fuchs and Dominique Bechmann and Yves Bertrand and Jean-Francois Dufourd. Formal Specification for Free-Form Curves and Surfaces, 12th Spring Conference on
Computer Graphics, pp. 121-130 (June 1996). Comenius University, Bratislava, Slovakia. Edited by Werner Purgathofer. ISBN 80-223-1032-8.
Suggested/Internal Citation Key
Fuchs:1996:FSF
#204:
Josef Hoschek. Dual Bézier curves and surfaces, Surfaces in Computer-Aided Geometric Design, pp. 147-156 (1983). North-Holland. Edited by Robert E. Barnhill and
Wolfgang Boehm.
Suggested/Internal Citation Key
Hoschek:1983:DBC
#205:
Robert E. Barnhill and Richard F. Riesenfeld. Computer Aided Geometric Design, (1974). Academic Press.
Keyword(s)
splines, curves and surfaces, design and modeling, CAD/ CAM
Suggested/Internal Citation Key
Barnhill:1974:CAG
#206:
Pierre E. Bezier. Emploi des machines a commande numerique, (1970). Masson et Cie..
Keyword(s)
splines, curves and surfaces, design and modeling, appl ications, CAD/CAM
Abstract
Translated by Forrest, A. Robin and Pankhurst, Anne F. as Numerical Control -- Mathematics and Applications, John Wiley and Sons, Ltd., London, 1972.
Suggested/Internal Citation Key
Bezier:1970:EDM
#207:
Steven A. Coons. Surfaces for Computer Aided Design, (1964). Design Division, Mech. Engin. Dept., M.I.T., Cambridge, Massachusetts.
Keyword(s)
curves and surfaces, design and modeling, CAD/CAM
Suggested/Internal Citation Key
Coons:1964:SFC
#208:
A. Robin Forrest. Curves and Surfaces for Computer-Aided Design, (July 1968). Cambridge University CAD Group.
Suggested/Internal Citation Key
Forrest:1968:CAS
#209:
William J. Gordon and Richard F. Riesenfeld. Bernstein-Bezier Methods for the Computer-Aided Design of Free-Form Curves and Surfaces, JACM, 21 (2), pp.
293-310 (April 1974).
Keyword(s)
spline, patch
Suggested/Internal Citation Key
Gordon:1974:BMF
#210:
W. Gordon and R. Riesenfeld. B-spline curves and surfaces, Computer Aided Geometric Design, pp. 95-126 (1974). Academic Press. Edited by R. E. Barnhill and R. F.
Riesenfeld.
Suggested/Internal Citation Key
Gordon:1974:BCA
#211:
Theodore M. P. Lee. Three-Dimensional Curves and Surfaces for Rapid Computer Display, (April 1969). Harvard U..
Suggested/Internal Citation Key
Lee:1969:TCA
#212:
A. Overhauser. Analytic definition of curves and surfaces by parabolic blending, (1968). Ford Motor Company.
Suggested/Internal Citation Key
Overhauser:1968:ADO
#213:
G. Salmon. A Treatise on Conic Sections, (1879). Longmans, Green, and Co..
Keyword(s)
conics, curves and surfaces
Suggested/Internal Citation Key
Salmon:1879:ATO
#214:
H. Spaeth. Spline algorithms for curves and surfaces, (1974). Utilitas Math. Publ. Inc., Winnipeg, Manitoba.
Abstract
From the German 'Spline Algorithmen zur Konstruktion glatter Kurven und Flaechen, R. Oldenburg Verlag, Muenchen'
Suggested/Internal Citation Key
Spaeth:1974:SAF
#215:
P. Y. Woon. On the Computer Drawing of Solid Objects Bounded by Quadric Surfaces, (June 1969). Department of Computer Science, New York University.
Keyword(s)
solids and volumes, curves and surfaces, graphics, algorithms
Suggested/Internal Citation Key
Woon:1969:OTC
#216:
P. Y. Woon. A Computer Procedure for Generating Visible Line Drawings for Solids Bounded by Quadric Surfaces, (December 1970). New York University.
Keyword(s)
solids and volumes, curves and surfaces, graphics, algorithms
Suggested/Internal Citation Key
Woon:1970:ACP
#217:
P. Y. Woon and H. Freeman. A Computer Procedure for Generating Visible Line Drawings for Solids Bounded by Quadric Surfaces, Proceedings of the IFIP Congress,
Information Processing '71, pp. 1120-1125 (1971). North-Holland Publishing Company.
Keyword(s)
solids and volumes, curves and surfaces, graphics, algorithms
Suggested/Internal Citation Key
Woon:1971:ACP
#218:
Edwin E. Catmull. Computer Display of Curved Surfaces, Proceedings of the IEEE Conference on Computer Graphics, Pattern Recognition, and Data Structure, pp.
11-17 (May 1975).
Keyword(s)
curves and surfaces, design and modeling, graphics, algorithms
Suggested/Internal Citation Key
Catmull:1975:CDO
#219:
Kenneth J. Versprille. Computer-Aided Design Applications of the Rational B-spline Approximation Form, (February 1975). Syracuse University.
Keyword(s)
splines, curves and surfaces, design and modeling, CAD/CAM
Suggested/Internal Citation Key
Versprille:1975:CDA
#220:
W. Boehm. Darstellung Und Korrektur Symmetrischer Kurven Und Flaechen Auf EDV Anlagen (Representation and Correction of Symmetric Curves and
Surfaces), Computing, 17 (), pp. 79-85 (1976).
Keyword(s)
curve correction and curve representation and symmetry
Suggested/Internal Citation Key
Boehm:1976:DUK
#221:
M. do Carmo. Differential Geometry of Curves and Surfaces, (1976). Prentice Hall.
Suggested/Internal Citation Key
Carmo:1976:DGO
#222:
J. M. Duncan. Application of differential geometry to computer curves and surfaces, (1976, Durham, England). University of Durham.
Keyword(s)
differential geometry
Suggested/Internal Citation Key
Duncan:1976:AOD
#223:
P. Bézier. Essay de définition numérique des courbes et des surfaces expérimentales, (1977). University of Paris VI.
Keyword(s)
curves and surfaces, design and modeling, CAD/CAM
Suggested/Internal Citation Key
Bezier:1977:EDD
#224:
W. Boehm. Cubic B-Spline Curves and Surfaces in Computer-Aided Geometric Design, Computing, 19 (), pp. 29-34 (1977).
Suggested/Internal Citation Key
Boehm:1977:CBC
#225:
P. J. Hartley and C. J. Judd. Parameterization of Bezier-Type B-Spline Curves and Surfaces, Computer-Aided Design, 10 (), pp. 130-134 (March 1978).
Keyword(s)
parametric and splines
Suggested/Internal Citation Key
Hartley:1978:POB
#226:
M. Hosaka and F. Kimura. Synthesis Methods of Curves and Surfaces in Interactive CAD, Proc. Interactive Techniques in Computer Aided Design (Bologna), pp.
151-156 (September 1978). IEEE Computer Society.
Keyword(s)
synthesis and interactive computer aided design
Suggested/Internal Citation Key
Hosaka:1978:SMO
#227:
Tony DeRose and Mary L. Bailey and Bill Barnard and Robert Cypher and David Dobbrikin and Carl Ebeling and Smaragda Konstantinidou and Larry McMurchie and Haim
Mizrahi and Bill Yost. Apex: two architectures for generating parametric curves and surfaces, The Visual Computer, 5 (5), pp. 264-276 (October 1989).
Keyword(s)
bezier, b-spline, cagd, hardware
Suggested/Internal Citation Key
DeRose:1989:ATA
#228:
Nicholas M. Patrikalakis and George A. Kriezis. Representation of piecewise continuous algebraic surfaces in terms of B-splines, The Visual Computer, 5 (6), pp.
360-374 (December 1989).
Keyword(s)
geometric modeling, algebraic curves and surfaces, b-splines, least squares
Suggested/Internal Citation Key
Patrikalakis:1989:ROP
#229:
Atsushi Yamada and Fujio Yamaguchi. Homogeneous bounding boxes as tools for intersection algorithms of rational bezier curves and surfaces, The Visual
Computer, 12(4), pp. 202-214 (1996). Springer-Verlag. ISSN 0178-2789.
Keyword(s)
CAD/CAM, computer graphics, rational Bézier curves and surfaces, intersection detection, projective spaces
Abstract
In the divide-and-conquer algorithm for detecting intersections of parametric rational Bézier curves (surfaces), we use bounding boxes in recursive rough checks. In this paper,
we replace the conventional bounding box with a homogeneous bounding box, which is projectively defined. We propose a new rough check algorithm based on it. One
characteristic of the homogeneous bounding box is that it contains a rational Bézier curve (surface) with weights of mixed signs. This replacement of the conventional bounding
box by the homogeneous one does not increase the computation time.
Suggested/Internal Citation Key
Yamada:1996:HBB
#230:
Shigeo Takahashi and Yoshihisa Shinagawa and Tosiyasu L. Kunii. Continuous-resolution-level constraints in variational design of multiresolution shapes, The Visual
Computer, 14(4), pp. 177-192 (1998). ISSN 0178-2789.
Keyword(s)
Continuous-resolution levels, Geometric constraints, Variational modeling, Curves and surfaces, Wavelets
Copyright
Copyright © 1998 Springer-Verlag
Abstract
This paper introduces continuous-resolution-level constraints to hierarchical editing of curves and surfaces based on B-spline wavelets. The constraints specify the shape at a
continuous-resolution level by interpolating those at integer-resolution levels. Energy functions subject to the shape deformations are used to control the smoothness of the
curves and surfaces. This paper proposes two interpolation schemes for the continuous-level shapes: linear interpolation and cardinal-spline interpolation. The
continuous-level shape is obtained as a transformation of that at an integer-resolution level, and the continuous-level constraints are reduced to those at integer-resolution levels.
Experimental results are presented to show that the continuous-level constraints effectively control the multiresolution curves and surfaces.
Suggested/Internal Citation Key
Takahashi:1998:CCI
#231:
Shigeo Takahashi. Variational design of curves and surfaces using multiresolution constraints, The Visual Computer, 14(5-6), pp. 208-227 (November 1998). ISSN
0178-2789.
Keyword(s)
Variational modeling, Multiresolution constraints, Smooth curves and surfaces, Endpoint-interpolating B-spline wavelets, Interactive 3D graphics
Copyright
Copyright © 1998 Springer-Verlag
Abstract
Variational design of curves and surfaces is a topic of interest in geometric modeling and interactive 3D graphics. Such variational methods have been extended to control
multiresolution curves and surfaces. However, in these methods, constraints imposed on the shape are common at all resolution levels; the level at which the shape satisfies
the constraints within the specified error tolerance is selected. We present a variational method of designing shapes that imposes different constraints at multiple levels of
resolution. The curves and surfaces are represented by endpoint-interpolating B-splines and their corresponding wavelets. Multiresolution constraints are converted from
coarse to fine resolution to associate all the constraints with common basis functions. We tested several combinations of energy functions and methods to see which is best for
controlling the smoothness.
Suggested/Internal Citation Key
Takahashi:1998:VDO
#232:
Tatiana Samoilov and Gershon Elber. Self-intersection elimination in metamorphosis of two-dimensional curves, The Visual Computer, 14(8-9), pp. 415-428 (1998). ISSN
0178-2789.
Keyword(s)
Computer-aided geometric design, Freeform parametric curves and surfaces, Homotopic curves and surfaces, Matching, Morphing
Copyright
Copyright © 1998 Springer-Verlag
Abstract
We consider two methods of self-intersection elimination in the metamorphosis of free-form planar curves. Both algorithms exploit a matching algorithm and construct the best
correspondence of the relative parameterizations of the initial and final curves. The first algorithm investigates building and employing a homotopy H:[0, 1]? R3 -> R3,
where H(t, r) for t=0 and t=1 are two given planar curves C1(r) and C2(r). The first t parameter defines the time of fixing the intermediate metamorphosis curve. The
locus of H(t, r) coincides with the ruled surface between C1(r) and C2(r), but each isoparametric curve of H(t, r) is self-intersection free. The second algorithm suits
morphing operations of planar curves. First, it constructs the best correspondence of the relative parameterizations of the initial and final curves. Then it eliminates the remaining
self-intersections and flips back the domains that self-intersect.
Suggested/Internal Citation Key
Samoilov:1998:SEI
#233:
Kauhuai Qin. General matrix representations for B-splines, The Visual Computer, 16 (3-4), pp. 177-186 (2000). ISSN 0178-2789.
Keyword(s)
B-splines, Matrix representations, Toeplitz matrix
Copyright
Copyright © 2000 Springer-Verlag
Abstract
In this paper, the concept of the basis matrix of B-splines is presented. A general matrix representation, which results in an explicitly recursive matrix formula, for nonuniform
B-spline curves of an arbitrary degree is also presented by means of the Toeplitz matrix. New recursive matrix representations for uniform B-spline curves and Bézier curves of
an arbitrary degree are obtained as special cases of that for nonuniform B-spline curves. The recursive formula for the basis matrix can be substituted for de Boor-Cox's formula
for B-splines, and it has a better time complexity than de Boor-Cox's formula when used for computation and conversion of B-spline curves and surfaces between different
CAD systems. Finally, some applications of the matrix representations are given in the paper.
Suggested/Internal Citation Key
Qin:2000:GMR
#234:
F. David Fracchia and Przemyslaw Prusinkiewicz. A physically-based finite element approach to modeling patches with n-sided polygonal domains, Proceedings of the 1991
Western Computer Graphics Symposium, pp. 21-25 (April 1991).
Keyword(s)
curves and surfaces
Suggested/Internal Citation Key
Fracchia:1991:APF
GRAPHBIB: Computer Graphics Bibliography Database