GRAPHBIB: Computer Graphics Bibliography Database

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

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

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

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

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

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

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

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

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

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

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

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

Gershon Elber. Metamorphosis of Free-form Curves and Surfaces, Computer Graphics International '95, (June 1995).

Suggested/Internal Citation Key
Elber:1995:MOF

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

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

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

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

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

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

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

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

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

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

C. Lin. Generalized Bernstein-Bézier curves and surfaces, Computer-Aided Design, 20 (5),  pp. 259-262 (1988).

Suggested/Internal Citation Key
Lin:1988:GBC

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

W. Boehm. Curvature continuous curves and surfaces, Computer Aided Geometric Design, 2 (2),  pp. 313-323 (1985).

Suggested/Internal Citation Key
Boehm:1985:CCC

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

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

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

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

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

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

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

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

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

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 R^d 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

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 BÈzier 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

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

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

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 ×n 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

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

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

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

 C-curves: An extension of cubic curves, 

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 G^n-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

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

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

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

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

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

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

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

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

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

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 C^n-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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

F. Yamaguchi. Curves and Surfaces in Computer Aided Geometric Design,  (1988). Springer.

Suggested/Internal Citation Key
Yamaguchi:1988:CAS

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A. Robin Forrest. Curves and Surfaces for Computer-Aided Design,  (July 1968). Cambridge University CAD Group.

Suggested/Internal Citation Key
Forrest:1968:CAS

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

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

Theodore M. P. Lee. Three-Dimensional Curves and Surfaces for Rapid Computer Display,  (April 1969). Harvard U..

Suggested/Internal Citation Key
Lee:1969:TCA

A. Overhauser. Analytic definition of curves and surfaces by parabolic blending,  (1968). Ford Motor Company.

Suggested/Internal Citation Key
Overhauser:1968:ADO

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

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

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

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

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

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

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

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

M. do Carmo. Differential Geometry of Curves and Surfaces,  (1976). Prentice Hall.

Suggested/Internal Citation Key
Carmo:1976:DGO

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

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

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

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

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

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

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

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

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

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

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

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

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