B-splines

-Bsplines

K. I. Joy. Utilizing parametric hyperpatch methods for modeling and display of free-form solids, Internat. J. Comput. Geom. Appl., 1 (4), pp. 455-471 (1991). Keyword(s) Bsplines, hypermatch models, free form surfaces, free form solids, sweeping operation Suggested/Internal Citation Key Joy:1991:UPH

Search Term: B-splines

#1:
Günther Greiner and Hans-Peter Seidel. Modeling with triangular B-splines, SMA '93: Proceedings of the Second Symposium on Solid Modeling and Applications, pp.
211-220 (May 1993, held May 19-21, 1993 in Montreal, Quebec, Canada). ACM.

Abstract
Triangular B-splines are a new tool for the modeling of complex objects with non-rectangular topology. The new B-spline scheme is based on blending functions and control points
and allows modeling piecewise polynomial surfaces over arbitrary triangulations with an optimal degree of smoothness. This paper discusses applications of triangular B-splines in
solid modeling. The new scheme is well-suited for modeling applications since it allows the construction of smooth surfaces with lowest algebraic degree possible. Furthermore, it is
possib]e to represent any piecewise polynomial surface as a linear combination of triangular B-splines. This triangular B-splines provide a unified data format for fairly arbitrary
surface types. Finally, triangular B-splines are ideally suited for blending applications.

Suggested/Internal Citation Key
Greiner:1993:MWT

#2:
Elaine Cohen and Tom Lyche and Larry L. Schumaker. Algorithms for degree raising of splines, ACM Transactions on Graphics, 4(3), pp. 171-181 (1985).

Abstract
Stable and efficient algorithms for degree-raising of curves (or surfaces) represented as arbitrary B-splines are presented as a application of the solution to the theoretical problem of
rewriting a curve written as a linear combination of mth order B-splines as a linear combination of (m + 1)st order B-splines with a minimal number of knot insertions. This
approach can be used to introduce additional degrees of freedom to a curve (or surface), a problem which naturally arises in certain circumstances in constructing mathematical models
for computer-aided geometric design.

Suggested/Internal Citation Key
Cohen:1985:AFD

#3:
Hartmut Prautzsch. A Round Trip to B-Splines Via de Casteljau, ACM Transactions on Graphics, 8 (3), pp. 243-254 (July 1989).

Suggested/Internal Citation Key
Prautzsch:1989:ART

#4:
A. A. Ball and D. J. T. Storry. An Investigation of Curvature Variations Over Recursively Generated B-Spline Surfaces, ACM Transactions on Graphics, 9(4), pp.
424-437 (October 1990). ISSN 0730-0301.

Keyword(s)
b-splines surfaces, curvature continuity, discrete Fourier transform, nonrectangular topologies, recursive subdivision

Abstract
The continuity properties of recursively generated B-spline surfaces over an arbitrary topology have been related to the eigenproperties of the local subdivision tranformation, and
conditions have been established on the subdivision weightings for tangent plane continuity at extraordinary points. In this paper, curves through an extradordinary point, which align
in both the tagent and binormal direction, are identified, and their curvatures are compared either side of the point. Further restrictions on the subdivision weightings are derived to
optimize the curvature properties of the surface. In general continuity of curvature is not attained.

Suggested/Internal Citation Key
Ball:1990:AIO

#5:
Alan Meyer. A Linear Time Oslo Algorithm, ACM Transactions on Graphics, 10(3), pp. 312-318 (July 1991). ISSN 0730-0301.

Keyword(s)
B-splines, computer-aided geometric design, Oslo algorithm, subdivision

Abstract
The Oslo algorithm provides a method for identifying and computing the nonzero discreti B-splines for a single new control point following a knot refinement. A linear time, finite
state, machine-baaed algorithm is presented which determines all discrete B-splines following a knot refinement. This can be viewed as a new front end to the Oslo algorithm.

Suggested/Internal Citation Key
Meyer:1991:ALT

#6:
S. L. Lee and A. A. Majid. Closed Smooth Piecewise Bicubic Surfaces, ACM Transactions on Graphics, 10(4), pp. 342-365 (October 1991). ISSN 0730-0301.

Keyword(s)
Bézier representation, bicubic patches, B-splines, closed surfaces, de Casteljau algorithm, geometric continuity, geometric modeling

Abstract
Two classes of closed piecewise bicubic surfaces are considered. These surfaces are geometrically smooth about each extraordinary point and, for a large mesh and a large control
polyhedron, they are parametrically smooth away from the extraordinary points. Furthermore, free parameters are available for manipulating the shape of the surface without changing
the control polyhedron. The control is local for a large control polyhedron.

Suggested/Internal Citation Key
Lee:1991:CSP

#7:
L. Ciminiera and P. Montuschi and A. Valenzano. A cellular array for computing bicubical B-splines coefficients, New Trends in Computer Graphics (Proceedings of CG
International '88), pp. 339-350 (1988). Springer-Verlag. Edited by N. Magnenat-Thalmann and D. Thalmann.

Suggested/Internal Citation Key
Ciminiera:1988:ACA

#8:
H. Huitric and M. Nahas. Realistic Effects with Rodin, Computer Graphics Tokyo '84, pp. T2-1/1-10 (1984).

Keyword(s)
B-Splines and B-Splines and Art

Suggested/Internal Citation Key
Huitric:1984:REW

#9:
G. Gallo and M. Spagnuolo. Uncertainty Coding and Controlled Data Reduction using Fuzzy-B-Splines, Computer Graphics International 1998, (June 1998, Hannover,
Germany). IEEE Computer Society.

Suggested/Internal Citation Key
Gallo:1998:UCA

#10:
Meleagros A. Krokos and Mel Slater. Interactive local control of interpolating curves using B(k-1)-splines, COMPUGRAPHICS '91, II (), pp. 182-191 (1991).

Keyword(s)
b-splines

Suggested/Internal Citation Key
Krokos:1991:ILC

#11:
L. Piegl and W. Tiller. Curve and surface constructions using rational B-splines, Computer-Aided Design, 19 (9), pp. 485-98 (1987).

Suggested/Internal Citation Key
Piegl:1987:CAS

#12:
B. Pham. Offset approximation of uniform B-splines, Computer Aided Design, 20 (8), pp. 471-474 (1988).

Suggested/Internal Citation Key
Pham:1988:OAO

#13:
L. Piegl. Modifying the shape of rational B-splines Part 1: curves, Computer Aided Design, 21 (8), pp. 509-518 (1989).

Suggested/Internal Citation Key
Piegl:1989:MTS

#14:
S. L. Lee and H. H. Tan and A. A. Majid. Smooth piecewise biquartic surfaces from quadrilateral control polyhedra with isolated n-sided faces, Computer-aided Design, 27(10),
pp. 741-758 (1995). Elsevier Science.

Keyword(s)
B-splines; Bézier representation; geometric modelling

Abstract
The modelling of surfaces by piecewise bipolynomial patches has important applications in computer-aided design and computer graphics. However, existing methods do not always
work for control polyhedra with arbitrary topology. A method is proposed that uses only piecewise biquartics which will work for control polyhedra with mostly quadrilateral faces
and isolated n-sided faces (n 4). The surfaces are modelled by converting the control points to Bézier points for each patch using a blending matrix. The blending matrices are
constructed from generalized biquartic B-spline blending functions. The domain of definition of these is a nonplanar rectangular surface mesh in 3D Euclidean space. The rectangular
mesh consists of rectangular faces whose vertices can have an arbitrary number of edges. The construction of the generalized biquartic B-spline blending functions is described. The
resulting surface is geometrically smooth around the extraordinary vertices and parametrically smooth otherwise. Furthermore, free parameters are available for manipulating the shape
of the surface locally around the extraordinary points.

Suggested/Internal Citation Key
Lee:1995:SPB

#15:
Jórg Peters. Biquartic C1-surface splines over irregular meshes, Computer-aided Design, 27(12), pp. 895-903 (1995). Elsevier Science.

Keyword(s)
C1 surface; corner cutting; tensor-product splines; spline mesh; blending; vertex-degree; polyhedral

Abstract
C1-surface splines define tangent continuous surfaces from control points in the manner of tensor-product (B-)splines, but allow a wider class of control meshes capable of
outlining arbitrary free-form surfaces with or without boundary. In particular, irregular meshes with non-quadrilateral cells and more or fewer than four cells meeting at a point can be
input and are treated in the same conceptual frame work as tensor-product B-splines; that is, the mesh points serve as control points of a smooth piecewise polynomial surface
representation that is local and evaluates by averaging. Biquartic surface splines extend and complement the definition of C1-surface splines in a previous paper (Peters, J SIAM J.
Numer. Anal. Vol 32 No 2 (1993) 645-666) improving continuity and shape properties in the case where the user chooses to model entirely with four-sided patches. While tangent
continuity is guaranteed, it is shown that no polynomial, symmetry-preserving construction with adjustable blends can guarantee its surfaces to lie in the local convex hull of the
control mesh for very sharp blends where three patches join. Biquartic C1-surface splines do as well as possible by guaranteeing the property whenever more than three patches
join and whenever the blend exceeds a certain small threshold.

Suggested/Internal Citation Key
Peters:1995:BS

#16:
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

#17:
TzuYi Yu and Bharat K. Soni. Application of NURBS in numerical grid generation, Computer-aided Design, 27(2), pp. 147-157 (1995). Elsevier Science.

Keyword(s)
NURBS; grid generation; computational fluid dynamics

Abstract
The application of nonuniform rational B-splines to grid generation involving complex industrial geometries is discussed. Geometrical entities are transformed and represented using
NURBS with common data structures. Techniques for reparameterization are discussed in terms of various grid topologies and distribution requirements. Computational examples
representative of practical computational fluid dynamics applications are presented to demonstrate the success of these methodologies.

Suggested/Internal Citation Key
Yu:1995:AON

#18:
Ioannis Fudos and Christoph M. Hoffmann. Constraint-based parametric conics for CAD, Computer-aided Design, 28(2), pp. 91-100 (1996). Elsevier Science.

Keyword(s)
conics for computer-aided design; geometric constraint solving; rational B-splines

Abstract
We describe how to construct conic blending arcs from constraints, using a unified rational parametric representation that combines the separate cases of blending parallel and
non-parallel edges. The possible constraints are that the arc must have a given distance from a line, a point, or a circle, or else intersect a circle or a line at a prescribed angle. Our
representation is easily converted into a rational B-spline with positive weights, and is therefore compatible with internal representations used by most solid modeling systems.
Finally, we discuss how we integrated this work with an algebraic constraint solver.

Suggested/Internal Citation Key
Fudos:1996:CPC

#19:
Hyungjun Park and Kwangsoo Kim. Smooth surface approximation to serial cross-sections, Computer-aided Design, 28(12), pp. 995-1005 (1996). Elsevier Science.

Keyword(s)
surface approximation; cross-sections; algorithms; B-splines; surface skinning

Abstract
The reconstruction of the surface model of an object from 2D cross-sections plays an important role in many applications. In this paper, we present a method for surface
approximation to a given set of 2D contours. The resulting surface is represented by a bicubic closed B-spline surface with C2 continuity. The method performs the skinning of
intermediate contour curves represented by cubic B-spline curves on a common knot vector, each of which is fitted to its contour points within a given accuracy. In order to acquire
more compact representation for the surface, the method includes an algorithm for reducing the number of knots in the common knot vector. The proposed method provides a smooth
and accurate surface model, yet realizes efficient data reduction. Some experimental results are given using synthetic and MRI data.

Suggested/Internal Citation Key
Park:1996:SSA

#20:
K. G. Pigounakis and P. D. Kaklis. Convexity-preserving fairing, Computer-aided Design, 28(12), pp. 981-994 (1996). Elsevier Science.

Keyword(s)
cubic curves; B-spline curves; convexity; tolerances; fairness; knot-removal; knot-insertion; curvature-slope discontinuity

Abstract
This paper develops a two-stage automatic algorithm for fairing C2-continuous cubic parametric B-splines under convexity, tolerance and end constraints. The first stage is a global
procedure, yielding a C2 cubic B-spline which satisfies the local- convexity, local-tolerance and end constraints imposed by the designer. The second stage is a local fine-fairing
procedure employing an iterative knot-removal knot-reinsertion technique, which adopts the curvature-slope discontinuity as the fairness measure of a C2 spline. This procedure
preserves the convexity and end properties of the output of the first stage and, moreover, it embodies a global-tolerance constraint. The performance of the algorithm is discussed for
four data sets.

Suggested/Internal Citation Key
Pigounakis:1996:CF

#21:
Der Min Tsay and Bor Jeng Lin. Improving the geometry design of cylindrical cams using nonparametric rational B-splines, Computer-aided Design, 28(1), pp.
5-15 (1996). Elsevier Science.

Keyword(s)
cylindrical cams; rational B-splines; interpolation

Abstract
A procedure for the synthesis and analysis of the surface geometry of cylindrical cams with oscillating roller followers is presented. The interpolation of nonparametric rational
B-splines is used to synthesize and refine the dwell-rise-dwell motion function of the follower. Given the synthesized follower motion, an approach based on the theory of
envelopes of a 1-parameter family of surfaces is used to determine the surface geometry of the cylindrical cam profile. To show the effect of the refined follower motion on the cam
surfaces, the geometric characteristics for the pressure angles and principal curvatures of the cylindrical cam surfaces are analysed. Favourable results obtained by using the rational
B-splines are illustrated in an application example, and they are compared with those obtained by other traditional methods.

Suggested/Internal Citation Key
Tsay:1996:ITG

#22:
Tzvetomir Ivanov Vassilev. Fair interpolation and approximation of B-splines by energy minimization and points insertion, Computer-aided Design, 28(9), pp.
753-760 (1996). Elsevier Science.

Keyword(s)
B-splines; interpolation; smoothing

Abstract
An efficient method for interpolation and approximation of both curve and surface points using B-splines is described. Automatic fairing is presented based on minimizing an energy
functional. Additional data points, used as degrees of freedom for the fairing, are inserted only where the curve (the surface) needs them. This reduces the number of the unknowns to
a minimum which makes the algorithm very fast and efficient especially when a huge amount of data is concerned. Results of applying the algorithm for about 15,000 face data
points, subject to measurement errors due to the digitization, are presented at the end of the paper.

Suggested/Internal Citation Key
Vassilev:1996:FIA

#23:
D. J. Walton and D. S. Meek. A triangular G1 patch from boundary curves, Computer-aided Design, 28(2), pp. 113-123 (1996). Elsevier Science.

Keyword(s)
triangular G1 patch; Gregory; Bézier

Abstract
For some applications it is necessary to fit an irregular surface to given data, e.g. to develop a geometric model of a human skeletal bone from computerized tomography scans. Such
a surface does not always have easily distinguishable isoparametric lines. It is thus not convenient to use standard global curve fitting techniques such as those based on B-splines.
A global method may also smooth away essential features. A reasonable approach is to use a composite surface where individual surface patches are locally determined. To obtain
some visual smoothness it is desirable that these patches join their neighbours in a manner that preserves positional as well as tangent plane continuity. Several methods have been
presented for constructing surfaces in such a manner. A common initial stage in developing the patches is to determine a network of boundary curves. This article reports on some
results using boundary curves based on a recent technique for point normal interpolation.

Suggested/Internal Citation Key
Walton:1996:AT

#24:
Xuefu Wang and Fuhua (Frank) Cheng and Brian A. Barsky. Energy and B-spline interproximation, Computer-aided Design, 29(7), pp. 485-496 (1997). Elsevier Science.

Keyword(s)
B-splines; interpolation; approximation; interproximation; non-linear programming; centripetal model; relative chord length parametrization; constrained optimization

Abstract
In this paper, we study B-spline curve interproximation with different energy forms and parametrization techniques, and present an interproximation scheme for B-spline surfaces. It
shows that the energy form has a much bigger impact on the generated curve than the parametrization technique. With the same energy form, different parametrization techniques
generate relatively small difference on the corresponding curves. With the same parametrization technique, however, different energy forms make significant difference on the shape
and smoothness of the resulting curves. Furthermore, interproximating B-spline curves generated by minimizing approximated energy forms are far from being good approximations
to the optimal curves. They tend to generate flatter regions and sharper turns than curves generated by minimizing the exact energy form. The interproximation scheme for surfaces is
aimed at generating a smooth surface to interpolate a grid of data which could either be a point or a region. This is achieved by minimizing a strain energy based on squared principal
curvatures for bicubic B-spline surfaces. The surface interproximation process is also studied with different energy forms and parametrization techniques. The test results of the
surface interproximation process also show the same conclusion as the curve interproximation process.

Suggested/Internal Citation Key
Wang:1997:EAB

#25:
S. T. Tuohy and T. Maekawa and G. Shen and N. M. Patrikalakis. Approximation of measured data with interval B-splines, Computer-aided Design, 29(11), pp.
791-799 (1997). Elsevier Science.

Keyword(s)
reverse engineering; robust solid modeling; interval methods; fitting

Abstract
The objective of this paper is to provide an efficient and reliable method for interpolating or approximating a set of measured data with an interval B-spline curve or surface. In
general, measured data possess uncertainty, arising from sensor precision and measurement registration, which can be represented as an interval. Both the interpolation and
approximation techniques presented in the paper produce interval bounding geometries that strictly enclose the intervals of the original data; in the case of our interpolation method, the
achieved fit is extremely tight, and in the case of our approximation technique the achieved fit depends on the number of control points one is willing to allow. Examples using
measured data illustrate our method.

Suggested/Internal Citation Key
Tuohy:1997:AOM

#26:
S. Guillet and J. C. Léon. Parametrically deformed free-form surfaces as part of a variational model, Computer-aided Design, 30(8), pp. 621-630 (1998). Elsevier Science.

Keyword(s)
free-form surfaces; B-splines; patched surfaces; deformations; constraints; optimization; interactive sculpting; bar network

Abstract
A new approach is described which provides deformation methods for multi-patch tensor based free-form surfaces. The surface deformation generated is controlled by global
geometric constraints. For example, the objective can be to deform a free-form surface until it becomes tangent to a pre-defined plane at a given point. This point can be fixed or free
to slide on the surface. The parametric deformation of surfaces is dedicated to modifications of free-form surfaces within CAD software and to the design of objects submitted to
aesthetic requirements. It is an alternative to previous approaches and it works with multiple surfaces through a simple mechanical model. The deformation method uses an analogy
between the control polyhedron of each surface (based on a B-Spline model) and the mechanical equilibrium of a rigid bar network. The user can localize the surface deformation into
an arbitrary shaped area through the selection of control polyhedron vertices spread over the entire surface. These vertices are used to automatically construct the associated bar
network. The bar network equilibrium parameters are set up to achieve isotropic or anisotropic deformation as required by the designer. The surface deformation is then automatically
carried through an optimization process which modifies mechanical parameters to agree with the global geometric constraint set up. The G1 continuity across the different during
the deformation process using a set of geometric constraints in addition to mechanical ones. Parametric free-form surface deformation can be subjected to non-linear geometric
constraints such as the tangency of a surface to a pre-defined plane. The resolution of such a problem uses an optimization process which minimizes the variation of the parameters
governing the equilibrium of the bar network, namely the external forces applied to the nodes of the network. Several examples illustrate basic deformation types with various sets of
constraints.

Suggested/Internal Citation Key
Guillet:1998:PDF

#27:
Weiyin Ma and Peiren He. B-spline surface local updating with unorganized points, Computer-aided Design, 30(11), pp. 853-862 (1998). Elsevier Science.

Keyword(s)
B-splines; shape modification; surface local updating; reverse engineering

Abstract
This paper presents an approach to update a local area of a B-spline surface based on a set of locally distributed and unorganized points in three-dimensional space. The region of the
original surface to be updated is first identified. The control points affecting these regions are further extracted or registered. The original B-spline surface is then updated by
modifying part or all of the registered control points through a local fitting process. Surface local updating can be done either with or without affecting the neighboring patches of the
region being updated. Depending on the number of patches involved in a region to be updated and whether the neighboring patches of a region are allowed for alteration, additional
knots may need to be inserted in the corresponding patches before surface local updating. If the shape of a region is likely to become more complex after updating, additional knots
may also need to be inserted in this region in order to obtain better fitting between the final updated surface and the locally distributed points. B-spline surface local updating is
particularly useful for free-form surface design and shape modification based on physical mockups.

Suggested/Internal Citation Key
Ma:1998:BSL

#28:
E. Saux and M. Daniel. Data reduction of polygonal curves using B-splines, Computer Aided Design, 31(8), pp. 507-515 (July 1999). Elsevier Science.

Keyword(s)
Data reduction; Accuracy criterion; B-splines; Smoothing

Abstract
We present a new method for data reduction of polygonal curves. Representation by means of a list of points does not provide fair curve models that may have complex and varying
shapes. We suggest a different technique based on fitting B-spline curves. This algorithm reaches high data reduction rates while producing fair approximations even for the most
complex curves. We apply our technique to cartographic data but the method is suitable for any application where the number of data points must be greatly reduced.

Suggested/Internal Citation Key
Saux:1999:DRO

#29:
C. Mandal and H. Qin and B. C. Vemuri. A novel FEM-based dynamic framework for subdivision surfaces, Computer-Aided Design, 32(8-9), pp. 479-497 (August 2000). ISSN
0010-4485.

Keyword(s)
Physics-based modeling, Geometric modeling, Computer graphics, CAGD, Subdivision surfaces, Deformable models, Dynamics, Finite elements, Interactive techniques

Abstract
Recursive subdivision on an initial control mesh generates a visually pleasing smooth surface in the limit. Nevertheless, users must carefully specify the initial mesh and/or
painstakingly manipulate the control vertices at different levels of subdivision hierarchy to satisfy a diverse set of functional requirements and aesthetic criteria in the limit shape. This
modeling drawback results from the lack of direct manipulation tools for the limit geometric shape. To improve the efficiency of interactive geometric modeling and engineering
design, in this paper we integrate novel physics-based modeling techniques with powerful geometric subdivision principles, and develop a unified finite element method (FEM)-based
methodology for arbitrary subdivision schemes. Strongly inspired by the recent research on Dynamic Non-Uniform Rational B-Splines (D-NURBS), we formulate and develop a
dynamic framework that permits users to directly manipulate the limit surface obtained from any subdivision procedure via simulated "force" tools. The most significant contribution
of our unified approach is the formulation of the limit surface of an arbitrary subdivision scheme as being composed of a single type of novel finite element. The specific geometric
and dynamic features of our subdivision-based finite elements depend on the subdivision scheme used. We present our novel FEM for the modified butterfly and Catmull-Clark
subdivision schemes, and generalize our dynamic framework to be applicable to other subdivision schemes. Our FEM-based approach significantly advances the state-of-the-art in
physics-based geometric modeling since it provides a universal physics-based framework for any subdivision scheme. In addition, we systematically devise a mechanism that allows
users to directly (not via control meshes) deform any subdivision surface; finally, we represent the limit surface of any subdivision scheme using a collection of subdivision-based
novel finite elements. Our experiments demonstrate that the new unified FEM-based framework not only promises a greater potential for subdivision techniques in solid modeling,
finite element analysis, and engineering design, but that it will further foster the applicability of subdivision geometry in a wide range of visual computing applications such as
visualization, virtual reality, computer graphics, computer vision, robotics, and medical imaging as well.

Suggested/Internal Citation Key
Mandal:2000:ANF

#30:
E. Lee. Some remarks concerning B-splines, Computer Aided Geometric Design, 2 (4), pp. 307-311 (1985).

Suggested/Internal Citation Key
Lee:1985:SRC

#31:
H. Seidel. A new multiaffine approach to B-splines, Computer Aided Geometric Design, 6 (1), pp. 23-32 (1989).

Suggested/Internal Citation Key
Seidel:1989:ANM

#32:
M. Mummy. Hermite interpolation with B-splines, Computer Aided Geometric Design, 6 (2), pp. 177-179 (1989).

Suggested/Internal Citation Key
Mummy:1989:HIW

#33:
S. Auerbach and R. Gmelig Meyling and M. Neamtu and H. Schaeben. Approximation and geometric modeling with simplex B-splines associated with irregular
triangles, Computer Aided Geometric Design, 8 (1), pp. 67-88 (1991).

Suggested/Internal Citation Key
Auerbach:1991:AAG

#34:
E. Lee and M. Lucian. Moebius reparametrizations of rational B-splines, Computer Aided Geometric Design, 8 (3), pp. 213-216 (1991).

Suggested/Internal Citation Key
Lee:1991:MRO

#35:
J. M. Carnicer and J. M. Peña. Totally positive bases for shape preserving curve design and optimality of B-splines, Computer Aided Geometric Design, 11(6), pp.
633-654 (1994). Elsevier Science. ISSN 0167-8396.

Keyword(s)
B-splines; Totally positive; Shape preserving; Normalized bases

Abstract
Normalized totally positive (NTP) bases present good shape preserving properties when they are used in Computer Aided Geometric Design. Here we characterize all the NTP bases
of a space and obtain a test to know if they exist. Furthermore, we construct the NTP basis with optimal shape preserving properties in the sense of (Goodman and Said, 1991), that
is, the shape of the control polygon of a curve with respect to the optimal basis resembles with the highest fidelity the shape of the curve among all the control polygons of the same
curve corresponding to NTP bases. In particular, this is the case of the B-spline basis in the space of polynomial splines. Further examples are given.

Suggested/Internal Citation Key
Carnicer:1994:TPB

#36:
E. T. Y. Lee. Computing a chain of blossoms, with application to products of splines, Computer Aided Geometric Design, 11(6), pp. 597-620 (1994). Elsevier Science. ISSN
0167-8396.

Keyword(s)
B-splines; Algorithms; Degree elevation; Product of splines; Blossoms; Raceme

Abstract
Instead of computing B-spline coefficients through the de Boor-Fix formula once for each coefficient, we derive a simple algorithm whereby groups of k coefficients are computed at
a time, k being the order of the spline. Besides other applications, our main objective is to give in detail a practical method for determining the B-spline representation of a product of
two splines.

Suggested/Internal Citation Key
Lee:1994:CAC

#37:
Phillip J. Barry and Dongli Su. Extending B-spline tools and algorithms to geometrically continuous splines: A study of similarities and differences, Computer Aided Geometric
Design, 12(6), pp. 581-600 (1995). Elsevier Science. ISSN 0167-8396.

Keyword(s)
B-spline; de Boor-fix dual functional; Discrete spline; Connection matrix; Evaluation; Geometric continuity; Knot insertion; Progressive curve; Total positivity

Abstract
This paper continues the exploration of geometrically continuous splines begun in (Dyn and Micchelli, 1988; Barry et al., 1991; Barry et al., 1993). Here we consider the question "to
what extent are the fundamental tools and algorithms derived for arbitrary degree geometrically continuous splines in (Seidel, 1993; Barry et al., 1993) similar to the tools and
algorithms for B-spline curves, and to what extent are they different?" To explore this question we present new results in four areas -- (i) explicit formulas for dual functionals for
geometrically continuous B-splines, (ii) complexity of the combinations in the algorithms, (iii) recurrences induced by these algorithms, and (iv) progressive curves in the
geometrically continuous setting. Each of these areas illustrates the similarities and differences between the tools and algorithms for geometrically continuous splines and tools and
algorithms for B-spline curves.

Suggested/Internal Citation Key
Barry:1995:EBT

#38:
Werner Hohenberger and Thomas Reuding. Smoothing rational B-spline curves using the weights in an optimization procedure, Computer Aided Geometric Design, 12(8), pp.
837-848 (1995). Elsevier Science. ISSN 0167-8396.

Keyword(s)
Smoothing; Optimization; Rational B-spline curves; NURBS

Abstract
Freeform shape design is typically accomplished in an interactive manner and shapes generated by a computer are rarely immediately acceptable. The available techniques for any
subsequent modifications depend on the chosen representation for the geometry. In many computer aided styling and design systems which use nonuniform rational B-splines
(NURBS) for representation of geometry, the use of the weights as a shape control tool is very inadequately supported. In fact they are often hidden from the user and therefore
remain unused. This paper investigates the possibilities of entering the weights in an automatic fairing process. In order to produce a curve with a more gradual change in curvature
and the smallest deviation from its initial shape the perturbation of the weights is stated as an optimization problem. Examples of applications to automotive shape design are presented
and discussed.

Suggested/Internal Citation Key
Hohenberger:1995:SRB

#39:
E. T. Y. Lee. Marsden's identity, Computer Aided Geometric Design, 13(4), pp. 287-305 (1996). Elsevier Science. ISSN 0167-8396.

Keyword(s)
Marsden's identity; Divided difference; Recurrence relations; de Boor-Fix formula; Blossoms; Rational linear transformations; Smoothness; Knot insertion; Degree elevation

Abstract
This paper consists of two parts. In the first part, a short new proof of Marsden's identity is given. It is shown that this identity is equivalent to the blossoming theorem, to a formula
for rational linear transformation of B-splines (and later, to the B-spline recurrence relations). In the second part, we develop the basic univariate spline theory entirely from
Marsden's identity (or rather, a preliminary form of it), without resorting to the divided difference machinery.

Suggested/Internal Citation Key
Lee:1996:MI

#40:
Wilfried Trump and Hartmut Prautzsch. Arbitrarily high degree elevation of Bézier representations, Computer Aided Geometric Design, 13(5), pp. 387-398 (1996). Elsevier
Science. ISSN 0167-8396.

Keyword(s)
Bézier curves; Bézier triangles; Bézier simplices; B-splines; Repeated degree elevation; Knot insertion; Simplicial recursions; Pyramidal schemes; Convergence

Abstract
In this paper we present fast algorithms to raise the degree n of a simplicial Bézier representation of degree n to arbitrarily high degree. Each Bézier point of some (n+r)th degree
representation can be computed in a simplicial recursive scheme of depth n . In the case of curves the recurrence relation reveals that the (n+r)th degree Bézier polygon can also be
obtained by inserting r knots into some nth degree spline which provides a very fast algorithm. Furthermore, a short new proof is given for the fact that the Bézier nets of a
multivariate polynomial converge to the polynomial under repeated degree elevation.

Suggested/Internal Citation Key
Trump:1996:AHD

#41:
Jiwen Zhang. Two different forms of C-B-splines, Computer Aided Geometric Design, 14(1), pp. 31-41 (1997). Elsevier Science. ISSN 0167-8396.

Keyword(s)
C-B-splines; Cubic uniform B-splines; C-curves; Nonuniform rational B-splines (NURBS)

Abstract
In a recent paper (Zhang, 1996), C-B-splines are introduced as extensions of cubic uniform B-splines. A new reparametrized form of C-B-splines, which is defined on the
interval [0,1], is proposed here. From this form, a third form that could have different parameters >alpha< in a curve is derived. These new forms give an efficient algorithm for
C-B-splines with any parameter >alpha< (0 &

Suggested/Internal Citation Key
Zhang:1997:TDF

#42:
Two different forms of C-B-splines,

#43:
Paul Dierckx. On calculating normalized Powell-Sabin B-splines, Computer Aided Geometric Design, 15(1), pp. 61-78 (1997). Elsevier Science. ISSN 0167-8396.

Keyword(s)
Linear programming; Quadratic programming; Powell-Sabin splines; Normalized B-splines; Control points; Bézier net

Abstract
An algorithm is presented for calculating a suitable normalized B-spline representation for Powell-Sabin splines in which the basis functions are all positive, have local support and
form a partition of unity. Computationally, the problem is reduced to the solution of a number of linear or quadratic programming problems of small size. Geometrically, each of these
can be interpreted as a problem of determining a triangle of minimal area, containing a specific subset of Bézier points. We further consider a number of CAGD applications such as
the determination of a suitable set of tangent control triangles and the efficient and stable calculation of the Bézier net of the PS-spline surface.

Suggested/Internal Citation Key
Dierckx:1997:OCN

#44:
Giulio Casciola. A recurrence relation for rational B-splines, Computer Aided Geometric Design, 14(2), pp. 103-110 (1997). Elsevier Science. ISSN 0167-8396.

Keyword(s)
Rational B-splines; Recurrence relation

Abstract
In this note a recurrence relation for rational B-splines is presented. Using this formula, it is possible to write a rational spline in terms of normalized rational B-splines of lower
order, with certain rational coefficients; these coincide with that generated by the well-known "rational version of the de Boor algorithm" based on knot- insertion (Farin, 1988 and
1989). A modified relation is presented that puts forward a different recurrence scheme.

Suggested/Internal Citation Key
Casciola:1997:ARR

#45:
Hong Qin and Demetri Terzopoulos. Triangular NURBS and their dynamic generalizations, Computer Aided Geometric Design, 14(4), pp. 325-347 (1997). Elsevier
Science. ISSN 0167-8396.

Keyword(s)
CAGD; Physics-based modeling; Triangular NURBS; Triangular B-splines; Dynamics; Constraints; Finite elements; Solid rounding; Scattered data fitting; Interactive sculpting

Abstract
Triangular B-splines are a new tool for modeling a broad class of objects defined over arbitrary, nonrectangular domains. They provide an elegant and unified representation scheme
for all piecewise continuous polynomial surfaces over planar triangulations. To enhance the power of this model, we propose triangular NURBS, the rational generalization of
triangular B-splines, with weights as additional degrees of freedom. Fixing the weights to unity reduces triangular NURBS to triangular B-splines. Conventional geometric design
with triangular NURBS can be laborious, since the user must manually adjust the many control points and weights. To ameliorate the design process, we develop a new model based
on the elegant triangular NURBS geometry and principles of physical dynamics. Our model combines the geometric features of triangular NURBS with the demonstrated
conveniences of interaction within a physics-based framework. The dynamic behavior of the model results from the numerical integration of differential equations of motion that
govern the temporal evolution of control points and weights in response to applied forces and constraints. This results in physically meaningful hence highly intuitive shape variation.
We apply Lagrangian mechanics to formulate the equations of motion of dynamic triangular NURBS and finite element analysis to reduce these equations to efficient numerical
algorithms. We demonstrate several applications, including direct manipulation and interactive sculpting through force-based tools, the fitting of unorganized data, and solid rounding
with geometric and physical constraints.

Suggested/Internal Citation Key
Qin:1997:TNA

#46:
Wayne Liu. A simple, efficient degree raising algorithm for B-spline curves, Computer Aided Geometric Design, 14(7), pp. 693-698 (1997). Elsevier Science. ISSN 0167-8396.

Keyword(s)
B-splines; Splines; Degree elevation; Algorithm; Blossoming

Abstract
This paper presents a new algorithm to compute the degree-raised version of a spline. The new algorithm is as fast as the best existing algorithm, but is much easier to understand and
to implement. The new control vertices of the degree-raised spline are obtained simply by a series of knot insertions followed by a series of knot deletions.

Suggested/Internal Citation Key
Liu:1997:ASE

#47:
Praveen Kashyap. Geometric interpretation of continuity over triangular domains, Computer Aided Geometric Design, 15(8), pp. 773-786 (1998). Elsevier Science. ISSN
0167-8396.

Abstract
In this paper, the geometric interpretation of higher order continuity conditions is presented. The concept of duality of "Bézier points" and "extension points" is developed by
considering different domain triangulation. Subsequently, the notion of "Continuity around a vertex" is realized and it results in some very interesting and symmetric geometries.
These constructs may motivate the geometric interpretation of triangular B-splines or triangular NURBS.

Suggested/Internal Citation Key
Kashyap:1998:GIO

#48:
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

#49:
Joris Windmolders and Paul Dierckx. Subdivision of uniform Powell-Sabin splines, Computer Aided Geometric Design, 16(4), pp. 301-315 (1999). Elsevier Science. ISSN
0167-8396.

Keyword(s)
Powell-Sabin splines; Subdivision; Wireframe; Normalized B-splines; Control points; Bézier net

Abstract
We propose a subdivision scheme for Powell--Sabin splines on uniform triangulations in their normalized B-spline representation. As an application we give an efficient algorithm for
displaying the surface.

Suggested/Internal Citation Key
Windmolders:1999:SOU

#50:
Choong-Gyoo Lim. A universal parametrization in B-spline curve and surface interpolation, Computer Aided Geometric Design, 16(5), pp. 407-422 (1999). Elsevier
Science. ISSN 0167-8396.

Keyword(s)
B-spline; Interpolation; Parametrization; Knot vector selection; Affine invariant; Transformation invariant

Abstract
We propose here a new universal parametrization for B-spline interpolation. The new parametrization is based on the values ti where B-splines Ni,k(t) are maximum in case of
order k. The resulting interpolation curve X(t) is transformation invariant and more natural looking, in general, than those obtained by other methods. Using a fixed knot vector, ti's
are independent of interpolating points {Pi}, and hence the computation of X(t) can be done more efficiently. In addition, the new method works well in any order k.

Suggested/Internal Citation Key
Lim:1999:AUP

#51:
B. I. Kvasov. Algorithms for shape preserving local approximation with automatic selection of tension parameters, Computer Aided Geometric Design, 17(1), pp. 17-37 (January
2000). ISSN 0167-8396.

Keyword(s)
Interval data, GB-splines, Shape preserving local approximation, Tensor product surfaces

Abstract
This paper describes the problem of shape preserving approximation for data with specified tolerances. Using the tool of generalized B-splines (GB-splines for short), simple one-
and three-point algorithms of shape preserving local approximation with automatic choice of the tension parameters are developed. In the two-dimensional case, tensor product of
one-dimensional splines are employed. The results of numerical calculations are given.

Suggested/Internal Citation Key
Kvasov:2000:AFS

#52:
Klaus Höllig. Stability of the B-spline basis via knot insertion, Computer Aided Geometric Design, 17 (5), pp. 447-450 (May 2000). ISSN 0167-8396.

Keyword(s)
B-splines, Knot insertion, Stability

Suggested/Internal Citation Key
Hollig:2000:SOT

#53:
Ulrich Reif. Best bounds on the approximation of polynomials and splines by their control structure, Computer Aided Geometric Design, 17(6), pp. 579-589 (July 2000). ISSN
0167-8396.

Keyword(s)
Best constant, Local bounds, Global bounds, Bézier curves, Bézier triangles, B-splines, Tensor product B-splines

Abstract
We present best bounds on the deviation between univariate polynomials, tensor product polynomials, Bézier triangles, univariate splines, and tensor product splines and the
corresponding control polygons and nets. Both pointwise estimates and bounds on the Lp-norm are given in terms of the maximum of second differences of the control points. The
given estimates are sharp for control points corresponding to arbitrary quadratic polynomials in the univariate case, and to special quadratic polynomials in the bivariate case.

Suggested/Internal Citation Key
Reif:2000:BBO

#54:
Michael Franssen and Remco C. Veltkamp and Wieger Wesselink. Efficient evaluation of triangular B-spline surfaces, Computer Aided Geometric Design, 17(9), pp.
863-877 (October 2000). ISSN 0167-8396.

Keyword(s)
Evaluation algorithms, Triangular B-splines

Abstract
Evaluation routines are essential for any application that uses triangular B-spline surfaces. This paper describes an algorithm to efficiently evaluate triangular B-spline surfaces with
arbitrary many variables. The novelty of the algorithm is its generality: there is no restriction on the degree of the B-spline surfaces or on the dimension of the domain. Constructing
an evaluation graph allows us to reuse partial results and hence, to decrease computation time. Computation time gets reduced even more by making choices in unfolding the
recurrence relation of simplex splines such that the evaluation graph becomes smaller. The complexity of the algorithm is measured by the number of leaves of the graph.

Suggested/Internal Citation Key
Franssen:2000:EEO

#55:
Ron Pfeifle and Hans-Peter Seidel. Spherical Triangular B-Splines with Application to Data Fitting, Computer Graphics Forum, 14(3), pp. 89-96 (August 1995). Blackwell
Publishers. Edited by Frits Post and Martin Göbel. ISSN 1067-7055.

Keyword(s)
spherical basis functions, DMS splines, triangular B-splines, scattered data fitting, affine coordinates

Abstract
Triangular B-splines surfaces are a tool for representing arbitrary piecewise polynomial surfaces over planar triangulations, while automatically maintaining continuity properties
across patch boundaries. Recently, Alfeld et al. [1] introduced the concept of spherical barycentric coordinates which allowed them to rmulate Bernstein-Bézier polynomials over the
sphere. In this paper we use the concept of spherical barycentric coordinates to develop a similar formulation for triangular B-splines, which we call spherical triangular B-splines.
These splines defined over spherical triangulations share the same continuity properties and similar evaluation algorithms with their planar counterparts, but possess none of the
annoying degeneracies found when trying to represent closed surfaces using planar parametric surfaces. We also present an example showing the use of these splines for
approximating spherical scattered data. Proceedings of Eurographics '95.

Suggested/Internal Citation Key
Pfeifle:1995:STB

#56:
Yizhou Yu and Qunsheng Peng. Multiresolution B-spline Radiosity, Computer Graphics Forum, 14(3), pp. 285-298 (August 1995). Blackwell Publishers. Edited by Frits Post
and Martin Göbel. ISSN 1067-7055.

Keyword(s)
radiosity, wavelet, B-spline

Abstract
This paper introduces a kind of new wavelet radiosity method called multiresolution B-spline radiosity, which uses B-splines of different scales to represent radiosity distribution
functions. A set of techniques and algorithms, such as function extrapolation, adaptive quadrature, scale adjustment and octree, are proposed to implement it. This method sets up
hierarchical structures on surfaces, keeps radiosity distribution continuous at element boundaries, does not need postprocessing, and does not prevent the use of any surface whose
parameter domain is rectilinear. Proceedings of Eurographics '95.

Suggested/Internal Citation Key
Yu:1995:MBR

#57:
Ron Pfeifle and Hans-Peter Seidel. Fitting Triangular B-Splines to Functional Scattered Data, Computer Graphics Forum, 15(1), pp. 15-24 (1996). ISSN 0167-7055.

Keyword(s)
scattered data approximation, triangular B-Splines, DMS splines, simplex splines, quadtrees

Abstract
Scattered data is, by definition, irregularly spaced. Uniform surface schemes are not well adapted to the locally varying nature of such data. Conversely, Triangular B-Spline surfaces
are more flexible in that they can be built over arbitrary triangulations and thus can be adapted to the scattered data. This paper discusses the use of DMS spline surfaces for
approximation of scattered data. A method is provided for automatically triangulating the domain containing the points and generating basis functions over this triangulation. A surface
approximating the data is then found by a combination of least squares and bending energy minimization. This combination serves both to generate a smooth surface and to
accommodate for gaps in the data. Examples are presented which demonstrate the effectiveness of the technique for mathematical, geographical and other data sets.

Suggested/Internal Citation Key
Pfeifle:1996:FTB

#58:
Michael D. McCool. Accelerated Evaluation of Box Splines via a Parallel Inverse FFT, Computer Graphics Forum, 15(1), pp. 35-46 (1996). ISSN 0167-7055.

Keyword(s)
box spline evaluation, volume rendering, Fast Fourier Transform, FFT, parallelism

Abstract
Box splines are a multivariate extension of uniform univariate B-splines. Direct evaluation of a box spline basis function can be difficult, but they have a relatively simple Fourier
transform and can therefore be evaluated with an inverse FFT. Symmetry, recursive evaluation of the coefficients, and parallelization can be used to improve absolute performance. A
windowing function can also be used to reduce truncation artifacts. We explore all these options in the context of a high-performance parallel implementation. Our goal is the
provision of an empirical touchstone for the inverse FFT evaluation of box spline basis functions,for eventual application to forward projection (splat-based) volume rendering.

Suggested/Internal Citation Key
McCool:1996:AEO

#59:
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

#60:
T. I. Vassilev. Interactive Sculpting with Deformable Nonuniform B-Splines, Computer Graphics Forum, 16(4), pp. 191-200 (1997). Blackwell Publishers. ISSN 1067-7055.

Keyword(s)
nonuniform B-splines, curve and surface energy minimization, interactive sculpting

Abstract
This paper describes an efficient method for manipulating deformable B-spline surfaces, based on minimizing an energy Functional. The major benefit of the proposed new fairness
norm is that it preserves the natural representation of the B-spline surface control points (a two dimensional array) which has an efficiency advantage over other methods. The
designer uses forces as a main sculpting tool and is free to specify a single force, a set of isolated forces, forces situated on a line or curve or area of the deformable surface. The user
is allowed to modify several parameters and in this way to change the physical properties of the object.

Suggested/Internal Citation Key
Vassilev:1997:ISW

#61:
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

#62:
Seungyong Lee. Interactive Multiresolution Editing of Arbitrary Meshes, Computer Graphics Forum, 18(3), pp. 73-82 (September 1999). Blackwell Publishers. ISSN 1067-7055.

Abstract
This paper presents a novel approach to multiresolution editing of a triangular mesh. The basic idea is to embed an editing area of a mesh onto a 2D rectangle and interpolate the
user-specified editing information over the 2D rectangle. The result of the interpolation is mapped back to the editing area and then used to update the mesh. We adopt harmonic maps
for the embedding and multilevel B-splines for the interpolation. The proposed mesh editing technique can handle an arbitrary mesh without any preprocessing such as remeshing. It
runs fast enough to support interactive editing and produces intuitive editing results.

Suggested/Internal Citation Key
Lee:1999:IME

#63:
Laurent Grisoni and Carole Blanc and Christophe Schlick. Hermitian B-Splines, Computer Graphics Forum, 18(4), pp. 237-248 (December 1999). Blackwell Publishers. ISSN
1067-7055.

Abstract
This paper proposes to study a spline model, called HB-splines, that is in fact a B-spline representation of Hermite splines, combined with some restriction on the differential values
at segment boundaries. Although this model does not appear able to offer something new to the computer graphics community, we think that HB-splines deserve to be considered
for themselves because they embed many interesting features. First, they include all the classical properties required in a geometric modeling environment (convex hull, local control,
arbitrary orders of parametric or geometric continuity). Second, they have a nice aptitude for direct manipulation (i.e. manipulation without using control points). For this purpose, we
propose a new graphic widget, called control sails, that offers the user an intuitive way to specify local properties (position, tangent, curvature) of a curve or a surface. Finally, they
provide an elegant formulation of a biorthogonal wavelet family, that permits multiresolution manipulations of the resulting curves or surfaces, in a very efficient way.

Suggested/Internal Citation Key
Grisoni:1999:HB

#64:
G. Creutz and C. Schubert. An interactive line creation method using B-splines, Computers & Graphics, 5 (2-4), pp. 71-78 (1980).

Keyword(s)
Algorithmic Aspects splines, line generation, Applications of Computer Graphics naval engineering, mechanical engineering general

Suggested/Internal Citation Key
Creutz:1980:AIL

#65:
M. Nahas and H. Huitric. Scenes of Our Imagination, Computers & Graphics, 7 (), pp. 205-207 (1983).

Keyword(s)
B-Splines and Interpolation Methods

Suggested/Internal Citation Key
Nahas:1983:SOO

#66:
Binh Pham. Quadratic B-splines for automatic curve and surface fitting, Computers & Graphics, 13 (4), pp. 471-475 (1989).

Suggested/Internal Citation Key
Pham:1989:QBF

#67:
Kai-Ching Chu. B3-splines for interactive curve and surface fitting, Computers & Graphics, 14 (2), pp. 281-288 (1990).

Keyword(s)
b-splines

Suggested/Internal Citation Key
Chu:1990:BFI

#68:
Karen M. Daniels and R. Daniel Bergeron and Georges G. Grinstein. Line monotonic partitioning of planar cubic B-splines, Computers & Graphics, 16 (1), pp. 55-68 (1992).

Suggested/Internal Citation Key
Daniels:1992:LMP

#69:
Shouqing Zhang and Ling Li and Hock Soon Seah. Vectorization of digital images using algebraic curves, Computers & Graphics, 22(1), pp. 91-101 (February 1998). Pergamon
Press / Elsevier Science. ISSN 0097-8493.

Keyword(s)
vectorization, curve fitting, algebraic curve, fine-tune and curve rendering

Abstract
An approach to vectorize digital curves is described that aims at representing a complicated scanned-in digital pictures in an analytical form using implicit algebraic curves for 2D
computer animation. An associated triangular quadtree visualization scheme is also presented in this paper to render the algebraic curve. Compared with curve fittings using parametric
methods, such as Bézier and B-Splines schemes, an algebraic curve with its implicit form should present more flexibility and ease the burden of geometrical reasonings and analytical
manipulations associated with parametric curve schemes.

Suggested/Internal Citation Key
Zhang:1998:VOD

#70:
J. Greissmair and W. Purgathofer. Deformation of Solids with Trivariate B-Splines, Eurographics '89, pp. 137-148 (1989). Eurographics. Edited by W. Hansmann and F. R. A.
Hopgoood and W. Strasser.

Suggested/Internal Citation Key
Greissmair:1989:DOS

#71:
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

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

#72:
Giovanni Gallo and Michela Spagnuolo and Salvatore Spinello. Fuzzy B-Splines: A Surface Model Encapsulating Uncertainty, Graphical Models, 62(1), pp. 40-55 (November
1999). Academic Press. ISSN 1524-0703.

Abstract
In the context of surface modeling, fuzzy B-splines are proposed as an integrated approach to uncertainty coding and data reduction. Fuzzy B-splines are suitable for representing
and simplifying both crisp and imprecise surface data and support interrogation of the model at different presumption levels. A high degree of compression can be achieved through a
procedure that defines the most significant representative among spatially clustered points. Experimental results are shown to prove the effectiveness of the proposed approach.

Suggested/Internal Citation Key
Gallo:2000:FBA

#73:
Ron Pfeifle and Hans-Peter Seidel. Fitting Triangular B-Splines to Functional Scattered Data, Graphics Interface '95, pp. 26-33 (May 1995). Canadian Human-Computer
Communications Society. Edited by Wayne A. Davis and Przemyslaw Prusinkiewicz. ISBN 0-9695338-4-5.

Keyword(s)
scattered data approximation, triangular B-splines, DMS splines, simplex splines, quadtree

Suggested/Internal Citation Key
Pfeifle:1995:FTB

#74:
Michael D. McCool. Optimized Evaluation of Box Splines via the Inverse FFT, Graphics Interface '95, pp. 34-43 (May 1995). Canadian Human-Computer Communications
Society. Edited by Wayne A. Davis and Przemyslaw Prusinkiewicz. ISBN 0-9695338-4-5.

Keyword(s)
box spline evaluation, B-splines, fast fourier transform (FFT), parallelism

Suggested/Internal Citation Key
McCool:1995:OEO

#75:
Ron Pfeifle and Hans-Peter Seidel. Triangular B-splines for Blending and Filling of Polygonal Holes, Graphics Interface '96, pp. 186-193 (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)
blending, filling polygonal holes, DMS splines, triangular b-splines, surface smoothing

Abstract
Triangular B-splines lend themselves naturally to problems such as blending and the filling of polygonal holes. Here we present an automatic method for smoothly blending
piecewise polynomial surfaces using triangular B-splines. The method proceeds in two phases. In the first phase, the domain of the region to be blended or filled is triangulated and
populated with basis functions. In the second phase, coefficients for these basis functions are found by minimizing a functional that measures the curvature of the blending or filling
surface. Examples are provided that show the use of this method for a number of blending and filling problems.

Suggested/Internal Citation Key
Pfeifle:1996:TBF

#76:
David Forsey and David Wong. Multiresolution Surface Reconstruction for Hierarchical B-splines, Graphics Interface '98, pp. 57-64 (June 1998). Edited by Kellogg Booth and
Alain Fournier.

Abstract
ISBN 0-9695338-6-1

Suggested/Internal Citation Key
Forsey:1998:MSR

#77:
D. F. Rogers and S. G. Satterfield and F. A. Rodriguez. Shiphulls, B-Spline Surfaces, and CAD/CAM, IEEE Computer Graphics & Applications, 3 (), pp. 37-45 (December
1983).

Keyword(s)
B-Splines and Mechanical Engineering

Suggested/Internal Citation Key
Rogers:1983:SBS

#78:
W. Tiller. Rational B-Splines for Curve and Surface Representation, IEEE Computer Graphics & Applications, 3 (), pp. 61-69 (1983).

Keyword(s)
Spline Curves and Spline Surfaces

Suggested/Internal Citation Key
Tiller:1983:RBF

#79:
H. Huitric and M. Nahas. B-Spline Surfaces: A Tool for Computer Painting, IEEE Computer Graphics & Applications, 5 (3), pp. 39-47 (March 1985).

Keyword(s)
splines (mathematics), paint systems, B-splines

Suggested/Internal Citation Key
Huitric:1985:BSA

#80:
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

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

#81:
Hong Qin and Demetri Terzopoulos. D-NURBS: A Physics-Based Framework for Geometric Design, IEEE Transactions on Visualization and Computer Graphics, 2(1), pp.
85-96 (March 1996). ISSN 1077-2626.

Keyword(s)
NURBS, geometric modeling, computer-aided design, computer graphics physics-based models, finite elements, dynamics

Abstract
This paper presents dynamic NURBS, or D-NURBS, a physics-based generalization of Non-Unifomm Rational B-Splines. NURBS have become a de facto standard in commercial
modellng systems because of their power to represent both free-form shapes and common analytic shapes. Traditionally, however, NURBS have been viewed as purely geometric
primitives, which require the designer to interactively adjust many degrees of freedom (DOFs - control points and associated weights - to achieve desired shapes. The conventional
shape modification process can often be clumsy and laborious. D-NURBS are physics-based models that incorporate mass distributions, internal deformation energies, forces, and
other physical quantities into the NURBS geometric substrate. Their dynamic behavior, resulting from the numerical integration of a set of nonlinear differential equations, produces
physically meaningful, hence intuitive shape variation. Consequently, a modeler can interactively sculpt complex shapes to required specifications not only in the traditional indirect
fashion, by adjusting control points and setting weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. We use
Lagrangian mechanics to fommulate the equations of motion for D-NURBS curves, tensor-product D-NURBS surfaces, swung D-NURBS surfaces, and triangular D-NURBS
surfaces. We apply finite element analysis to reduce these equations to eflficient numerical algorithms computable at interactive rates on common graphics workstations. We
implement a prototype modeling environment based on D-NURBS and demonstrate that D-NURBS can be effective tools in a wide range of CAGD applications such as shape
blending, scattered data fitting, and interactive sculpting.

Suggested/Internal Citation Key
Qin:1996:DAP

#82:
Seungyong Lee and George Wolberg and Sung Yong Shin. Scattered Data Interpolation with Multilevel B-Splines, IEEE Transactions on Visualization and Computer
Graphics, 3(3), pp. 228-244 (July - September 1997). ISSN 1077-2626.

Keyword(s)
Scattered data interpolation, multilevel B-splines, data approximation

Abstract
This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel B-splines are introduced to compute a C2-continuous surface through a set of
irregularly spaced points. The algorithm makes use of a coarse-to-fine hierarchy of control lattices to generate a sequence of bicubic B-spline functions whose sum approaches the
desired interpolation function. Large performance gains are realized by using B-spline refinement to reduce the sum of these functions into one equivalent B-spline function.
Experimental results demonstrate that high-fidelity reconstruction is possible from a selected set of sparse and irregular samples.

Suggested/Internal Citation Key
Lee:1997:SDI

#83:
Brian A. Barsky. A description and evaluation of various 3d models, Computer Graphics Theory and Applications (Proceedings of InterGraphics '83), pp. 75-95 (April
1983). Springer-Verlag. Edited by Tosiyasu L. Kunii.

Keyword(s)
hermite interpolation, coons surfaces, b-splines

Suggested/Internal Citation Key
Barsky:1983:ADA

#84:
Frederick W. B. Li and Rynson W. H. Lau. Incremental Polygonization of Deforming NURBS Surfaces, Journal of Graphics Tools, 4(4), pp. 37-50 (1999). ISSN 1086-7651.

Abstract
Nonuniform rational B-splines (NURBS) are a powerful tool to model deformable objects. Their shapes can be easily modified by moving the control points. A common method
used to render these objects is polygonization. However, the polygonization process is computationally very expensive. If the object deforms, we need to execute this process in
every frame to reflect the geometric change of the object. This limitation makes real-time rendering of deforming objects very difficult. In this paper, we present an incremental method
for polygonizing deforming objects modeled by NURBS surfaces. Some incremental techniques are introduced here to further improve the performance of the method. They include
an efficient mechanism for determining the deformation region when the surface deforms, an incremental crack prevention technique, and an updating method for multiple control
point movement.

Suggested/Internal Citation Key
Li:1999:IPO

#85:
S. T. Tuohy and N. M. Patrikalakis. Non-linear Data Representation for Ocean Exploration and Visualization, Journal of Visualization and Computer Animation, 7(3), pp.
135-140 (July - September 1996). ISSN 1049-8907.

Keyword(s)
spline approximation, geophysical maps, interval methods

Abstract
This paper proposes a method for the representation of functions describing a measured geophysical property (via sparsely scattered ordensely defined point data) by tensor and triple
product interval B-splines (IBS). The spline representation facilitates archiving, data storage reduction, visualization and more general high-level interrogation. Interval methods
allow for the representation of the function values together with their uncertainty. The uncertainty is introduced, for example, because the measurement (or dependent variable) or the
location of the sensor (the independent variable(s)) is known only to a finite precision. In this paper, we present algorithms for the creation of IBS geometries based on minimization
with linear constraints and we illustrate the method using geophysical ocean data and their interrogation.

Suggested/Internal Citation Key
Tuohy:1996:NDR

#86:
Jan Helge Bohn. Computing the unit normal for non-uniform rational B-splines surfaces, (December 1989). Rensselaer Polytechnic Institute.

Keyword(s)
nurbs

Abstract
Advisor: S. Abi-Ezzi and R. O'Bara and M. Wozny

Suggested/Internal Citation Key
Bohn:1989:CTU

#87:
S. Abi-Ezzi. The graphical processing of B-splines in a highly dynamic environment, (1989). RPI.

Abstract
Rensselaer Design Research Center

Suggested/Internal Citation Key
Abi-Ezzi:1989:TGP

#88:
E. Cohen and T. Lyche and R. Riesenfeld. Discrete B-Splines and Subdivision Techniques in Computer-Aided Geometric Design and Computer Graphics, Comput. Gr. Image
Process., 14 (), pp. 87-111 (October 1980).

Keyword(s)
Algorithmic Aspects splines and subdivision techniques

Suggested/Internal Citation Key
Cohen:1980:DBA

#89:
W. Dahmen. On multivariate B-splines, SIAM J Numerical Analysis, 17 (2), pp. 179-191 (1980).

Suggested/Internal Citation Key
Dahmen:1980:OMB

#90:
H. Huitric and M. Nahas and R. E. A. Mason. Computer Art with Rodin, Information Processing 83. Proceedings of the IFIP 9th World Computer Congress, pp. 275-282 (1983).

Keyword(s)
B-Splines and Realism and Art

Suggested/Internal Citation Key
Huitric:1983:CAW

#91:
P. Kochevar. An application of multivariate B-splines to computer- aided geometric design, Rocky Mtn. J of Math., 14 (1), pp. 159-175 (1984).

Suggested/Internal Citation Key
Kochevar:1984:AAO

#92:
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

#93:
K. MacCallum and J.-M. Zhang. Curve smoothing techniques using B-splines, The Computer J, 29 (6), (1986).

Suggested/Internal Citation Key
MacCallum:1986:CST

#94:
John W. Peterson. Ray Tracing General B-Splines, Proceedings of the ACM Mountain Regional Conference, pp. 87 (April 1986).

Keyword(s)
B-splines, surfaces

Abstract
extensions to Sweeney's patch algorithm to handle a wider range of surfaces

Suggested/Internal Citation Key
Peterson:1986:RTG

#95:
M. Sabin. Open questions in the application of multivariate B-splines, Mathematical Methods in Computer Aided Geometric Design, pp. 529-538 (1989). Academic Press. Edited
by T. Lyche and L. Schumaker.

Suggested/Internal Citation Key
Sabin:1989:OQI

#96:
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

#97:
Tapio Takala and Charles D. Woodward. Industrial design based on geometric intentions, Theoretical Foundations of Computer Graphics and CAD, NATO ASI, F40 (), pp.
953-963 (1988). Springer-Verlag. Edited by R. A. Earnshaw.

Keyword(s)
design theory, design transactions, geometric modeling, b-splines

Suggested/Internal Citation Key
Takala:1988:IDB

#98:
K. Qin. General Matrix Representations for B-Splines and Applications, Pacific Graphics '98, (October 1998, Singapore).

Suggested/Internal Citation Key
Qin:1998:GMR

#99:
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

Suggested/Internal Citation Key
Joe:1987:DB

#100:
Charles Loop. Smooth Spline Surfaces over Irregular Meshes, Proceedings of SIGGRAPH 94, Computer Graphics Proceedings, Annual Conference Series, pp. 303-310 (July
1994, Orlando, Florida). ACM Press. Edited by Andrew Glassner. ISBN 0-89791-667-0.

Keyword(s)
Computer-aided geometric design, B-spline surfaces, Triangular patches, Geometric continuity, Irregular meshes, Aribitrary topology

Abstract
An algorithm for creating smooth spline surfaces over irregular meshes is presented. The algorithm is a generalization of quadratic B-splines; that is, if a mesh is (locally) regular,
the resulting surface is equivalent to a B-spline. Otherwise, the resulting surface has a degree 3 or 4 parametric polynomial representation. A construction is given for representing the
surface as a collection of tangent plane continuous triangular Bézier patches. The algorithm is simple, efficient, and generates aesthetically pleasing shapes.

Suggested/Internal Citation Key
Loop:1994:SSS

#101:
Cindy M. Grimm and John F. Hughes. Modeling Surfaces of Arbitrary Topology using Manifolds, Proceedings of SIGGRAPH 95, Computer Graphics Proceedings, Annual
Conference Series, pp. 359-368 (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.

Abstract
We describe an extension of B-splines to surfaces of arbitrary topology, including arbitrary boundaries. The technique inherits many of the properties of B-splines: local control, a
compact representation, and guaranteed continuity of arbitrary degree. The surface is specified using a polyhedral control mesh instead of a rectangular one; the resulting surface
approximates the polyhedral mesh much as a B-spline approximates its rectangular control mesh. Like a B-spline, the surface is a single, continuous object. This is achieved by
modeling the domain of the surface with a manifold whose topology matches that of the polyhedral mesh, then embedding this domain into 3-space using a
basis-function/control-point formulation. We provide a constructive approach to building a manifold.

Suggested/Internal Citation Key
Grimm:1995:MSO

#102:
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.

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

#103:
Thomas W. Sederberg and Jianmin Zheng and David Sewell and Malcolm Sabin. Non-Uniform Recursive Subdivision Surfaces, Proceedings of SIGGRAPH 98, Computer
Graphics Proceedings, Annual Conference Series, pp. 387-394 (July 1998, Orlando, Florida). Addison Wesley. Edited by Michael Cohen. ISBN 0-89791-999-8.

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)
B-splines, Doo-Sabin surfaces, Catmull-Clark surfaces

Abstract
Sabin and Catmull-Clark subdivision surfaces are based on the notion of repeated knot insertion of uniform tensor product B-spline surfaces. This paper develops rules for
non-uniform Doo-Sabin and Clark surfaces that generalize non-uniform tensor product spline surfaces to arbitrary topologies. This added flexibility allows, among other things, the
natural introduction of features such as cusps, creases, and darts, while elsewhere maintaining the same order of tinuity as their uniform counterparts.

Suggested/Internal Citation Key
Sederberg:1998:NRS

This article references some others in the database:
Halstead:1993:EFI | Hoppe:1994:PSS | Loop:1994:SSS | Zorin:1996:ISM

#104:
Jos Stam. Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values, Proceedings of SIGGRAPH 98, Computer Graphics Proceedings, Annual
Conference Series, pp. 395-404 (July 1998, Orlando, Florida). Addison Wesley. Edited by Michael Cohen. ISBN 0-89791-999-8.

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)
subdivision surfaces, eigenanalysis, linear algebra, parametrizations, surface evaluation, Catmull-Clark surfaces

Abstract
In this paper we disprove the belief widespread within the computer graphics community that Catmull-Clark subdivision surfaces cannot be evaluated directly without explicitly
subdividing. We show that the surface and all its derivatives can be evaluated in terms of a set of eigenbasis functions which depend only on the subdivision scheme and we derive
analytical expressions for these basis functions. In particular, on the regular part of the control mesh where Catmull-Clark surfaces are bi-cubic B-splines, the eigenbasis is equal to
the power basis. Also, our technique is both easy to implement and efficient. We have used our implementation to compute high quality curvature plots of subdivision surfaces. The
cost of our evaluation scheme is comparable to that of a bicubic spline. Therefore, our method allows many algorithms developed for parametric surfaces to be applied to
Catmull-Clark subdivision surfaces. This makes subdivision surfaces an even more attractive tool for free-form surface modeling.

Suggested/Internal Citation Key
Stam:1998:EEO

This article references some others in the database:
Halstead:1993:EFI

#105:
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

#106:
M. Cox. The numerical evaluation of B-splines, (1971). National Physical Laboratory.

Suggested/Internal Citation Key
Cox:1971:TNE

#107:
M. Cox. The numerical evaluation of B-splines, J Inst. Maths. Applics., 10 (), pp. 134-149 (1972).

Suggested/Internal Citation Key
Cox:1972:TNE

#108:
T. Greville. On the normalization of the B-splines and the location of the nodes for the case of unequally spaced knots, Inequalities, (1967). Academic Press. Edited by O. Shisha.

Abstract
Supplement to the paper `On spline functions' by I. Schoenberg

Suggested/Internal Citation Key
Greville:1967:OTN

#109:
C. de Boor. On local linear functions which vanish at all B-splines but one, Theory of approximation with applications, pp. 120-145 (1976). Academic Press, New York. Edited
by A. Law and B. Sahney.

Suggested/Internal Citation Key
Boor:1976:OLL

#110:
C. de Boor. Splines as linear combinations of B-splines -- a survey, Approximation Theory II, pp. 1-47 (1976). Academic Press, New York. Edited by G. Lorentz and C. Chui
and L. Schumaker.

Suggested/Internal Citation Key
Boor:1976:SAL

#111:
C. de Boor. Package for calculating with B-splines, Journal of Numerical Analysis, 14 (3), pp. 441-472 (1977).

Suggested/Internal Citation Key
Boor:1977:PFC

#112:
T. Lyche and R. Winther. A stable recurrence relation for trigonometric B-splines, J. Approximation Theory, 25 (3), pp. 266-279 (1979).

Suggested/Internal Citation Key
Lyche:1979:ASR

#113:
F. C. Munchmeyer and C. Schubert and H. Nowacki. Interactive Design of Fair Hull Surfaces Using B-Splines, Comput. Industry, pp. 77-86 (December 1979).

Keyword(s)
Algorithmic Aspects splines and surface analysis and surface splines

Suggested/Internal Citation Key
Munchmeyer:1979:IDO

#114:
Charles D. Woodward. B2-splines: a local representation for cubic spline interpolation, The Visual Computer, 3 (3), pp. 152-161 (October 1987).

Keyword(s)
free-form modelling, interpolation, interaction, b-splines

Suggested/Internal Citation Key
Woodward:1987:BAL

#115:
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

#116:
Pei Zhao and Hung Chuan Teh. Rational bicubic simple quadrilateral mesh surfaces, The Visual Computer, 11(8), pp. 401-418 (1995). Springer-Verlag. ISSN 0178-2789.

Keyword(s)
Closed surface modeling, B-splines, Rational bicubic patches, Bézier representation, Free-form surfaces design

Abstract
In free-form modeling a closed smooth piecewise surface is highly desirable when the smoothness across the boundaries of patches can be represented within the formulation.
Closed, smooth, piecewise bicubic surfaces, defined on simple quadrilateral mesh. (SQM), may be defined as SQM surfaces. We have extended previous work on SQM surfaces and
described the surface representation in rational form. The global constraint of the control parameters associated with each control vertex is relaxed. The new local shape-control
parameters with their larger range of usability further enhance the power of this free-form surface design scheme. We have also provided more B-spline functions. A complete set of
B-spline functions for various topologies of the SQM is now available. Examples demonstrate that editing of shapes for reasonably complex objects can be carried out on an SGI
Personal Iris machine at an interactive rate.

Suggested/Internal Citation Key
Zhao:1995:RBS

#117:
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

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

#118:
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

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

GRAPHBIB: Computer Graphics Bibliography Database