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## Curves and Surfaces for CAGD (1993)

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Citations: | 76 - 1 self |

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1791 |
A practical guide to splines.
- Boor
- 1978
(Show Context)
Citation Context ...ade B-splines a truly viable tool in CAGD. Before its discovery, B-splines were defined using a tedious divided difference approach which was numerically very unstable. For a detailed discussion, see =-=[42]-=-. Spline functions are important in approximation theory, but in CAGD, parametric spline curves are much more important. These were introduced in 1974 by R. Riesenfeld and W. Gordon [73] (the paper is... |

1119 |
Curves and Surfaces for Computer Aided Geometric Design
- Farin
- 2002
(Show Context)
Citation Context ... identical to de Casteljau’s curves, only the mathematics involved was different. A member of Bézier’s team, D. Vernet independently developed the de Casteljau algorithm. See also Bézier’s chapter in =-=[59]-=-. 5s6 Figure 5. De Casteljau’s description of his algorithm. Bézier’s work was widely published, see [15,12–14,138], and soon came to the attention of A.R. Forrest. He realized that Bézier curves coul... |

737 |
Recursively generated B-spline surfaces on arbitrary topological meshes
- Catmull, Clark
- 1978
(Show Context)
Citation Context ...ial work on subdivision surfaces, going back to two articles in the no. 6 volume of the journal Computer Aided Design, 1978. They were authored by D. Doo and M. Sabin [50] and E. Catmull and J. Clark =-=[26]-=-. Both papers have a similar flavor. Chaikin’s algorithm can be generalized to tensor product surfaces in a straightforward way. Such surfaces have a rectilinear control mesh, and after analyzing the ... |

616 |
Smooth subdivision surfaces based on triangles
- LOOP
- 1987
(Show Context)
Citation Context ...urfaces have a piecewise biquadratic flavor; the Catmull/Clark ones have a piecewise bicubic flavor. In 1987, C. Loop generalized triangular spline surfaces to a new G 1 subdivision surface type, see =-=[96]-=-. Its input control polygon can be any triangular mesh. Loop’s algorithm is based on a subdivision scheme for so-called box-splines by W. Boehm [21] and H. Prautzsch [108]. All three of the above algo... |

393 | A Butterfly Subdivision Scheme for Surface Interpolation with Tension Control
- Dyn, Levin, et al.
- 1990
(Show Context)
Citation Context ...n discovered an interpolating subdivision scheme, called the 4-point curve scheme, see [54]. It was generalized to surfaces – to the so-called “butterfly algorithm” – by the same authors in 1990, see =-=[53]-=-. 9. Scientific Applications Many areas of science need to model phenomena for which only a set of discrete measurements is available – an example is a weather map where data are collected at a set of... |

340 |
Splines minimizing rotation-invariant seminorms in Sobolev spaces.
- Duchon
- 1977
(Show Context)
Citation Context ...is was given by C. Micchelli in 1986 at the Mathematics Research Center in Oberwolfach, Germany.sA different generalization of splines to surfaces is the method of thin plate splines due to J. Duchon =-=[52]-=-. Thin plate splines minimize an “energy functional”, similar to the one that spline curves minimize. While spline curves simulate the behavior of an elastic beam, thin plate splines minimize the beha... |

274 |
Multiquadric equations of topography and other irregular surfaces
- Hardy
- 1971
(Show Context)
Citation Context ...evaluation point. The method itself is too poorly behaved to be viable on its own, but it was used as an ingredient in other methods (see the above references). Another approach was taken by R. Hardy =-=[80]-=- in 1971, who generalized the concept of splines to surfaces. His surfaces utilize radial basis functions which are bell-shaped functions (reminiscent of univariate B-splines) with their maxima at the... |

273 |
Behaviour of recursive division surfaces near extraordinary points
- Doo, Sabin
- 1978
(Show Context)
Citation Context ...s the starting point for the initial work on subdivision surfaces, going back to two articles in the no. 6 volume of the journal Computer Aided Design, 1978. They were authored by D. Doo and M. Sabin =-=[50]-=- and E. Catmull and J. Clark [26]. Both papers have a similar flavor. Chaikin’s algorithm can be generalized to tensor product surfaces in a straightforward way. Such surfaces have a rectilinear contr... |

202 |
A four-point interpolatory subdivision scheme for curve design
- Dyn, Gregory, et al.
- 1987
(Show Context)
Citation Context .... Early work on subdivision surfaces focussed on approximating surfaces. In 1987, N. Dyn, J. Gregory, and D. Levin discovered an interpolating subdivision scheme, called the 4-point curve scheme, see =-=[54]-=-. It was generalized to surfaces – to the so-called “butterfly algorithm” – by the same authors in 1990, see [53]. 9. Scientific Applications Many areas of science need to model phenomena for which on... |

148 |
On calculating with B-splines
- Boor
- 1972
(Show Context)
Citation Context ...ation. He later became one of the most influential proponents of B-splines in approximation theory. The recursive evaluation of B-spline curves is due to him and is now known as the de Boor algorithm =-=[40]-=-. It is based on a recursion for B-splines which was independently discovered by de Boor, L. Mansfield, and M. Cox [37]. It was this recursion that made B-splines a truly viable tool in CAGD. Before i... |

142 |
Computing Dirichlet tessellations in the plane
- Green, Sibson
- 1978
(Show Context)
Citation Context ... a given 2D point set as vertices. The first algorithm was published in 1971 by C. Lawson [94]. An algorithmic connection between triangulations and Voronoi diagrams was presented by Green and Sibson =-=[76]-=-. The n−dimensional case was covered by D. Watson [140]. 2D triangulations are an important preprocessing tool for many surface fitting operations. 3D point sets arise when physical objects are digiti... |

111 |
Discrete B-splines and subdivision techniques in computeraided geometric design and computer graphics
- Cohen, Lyche, et al.
- 1980
(Show Context)
Citation Context ...s were soon developed that simplified the mathematical treatment of B-spline curves; these include Boehm’s knot insertion algorithm 6 [20], the Oslo algorithm by E. Cohen, T. Lyche, and R. Riesenfeld =-=[32]-=-, and the introduction of the blossoming principle by L. Ramshaw [110] and P. de Casteljau [45]. The generalization of B-spline curves to NURBS – Nonuniform rational B-splineS – has become the standar... |

105 |
An algorithm for high speed curve generation
- Chaikin
- 1974
(Show Context)
Citation Context ...did not fit the mold of any of the other methods of the conference. Starting from a closed 2D polygon, and using a process of continual “chopping off corners,” he arrived at a smooth limit curve, see =-=[27]-=-. At the conference, both R. Riesenfeld and M. Sabin argued that Chaikin had invented an iterative way to generate uniform quadratic B-spline curves, see [113]. In 1987, C. de Boor discovered that “co... |

86 |
Triangular Bernstein–Bézier patches
- Farin
- 1986
(Show Context)
Citation Context ...c elements [92] or C 1 split triangle cubic elements. The latter one, devised by Clough and Tocher as a finite element [31], gained some popularity in the context of scattered data interpolation, see =-=[58]-=-. From a historical perspective, it interesting to observe that early finite element research on triangular patches did not make use of the elegant formalism offered by the use of barycentric coordina... |

86 | The curve of least energy
- Horn
- 1983
(Show Context)
Citation Context ... an important element early on in the development of CAGD. Since interpolating spline curves were known to exhibit unwanted undulations, alternatives were studied, typically involving curvature – see =-=[99,100,82,125]-=-. A different approach is to take an existing curve or surface and inspect its shape: if it is imperfect, apply a fairing procedure. Such methods typically aim at removing noise from either data point... |

69 |
J.: Finite element stiffness matrices for the analysis of plate bending
- Clough, Tocher
- 1965
(Show Context)
Citation Context ...the very beginning of Ritz-Galerkin methods. Higher order elements use C 1 quintic elements [92] or C 1 split triangle cubic elements. The latter one, devised by Clough and Tocher as a finite element =-=[31]-=-, gained some popularity in the context of scattered data interpolation, see [58]. From a historical perspective, it interesting to observe that early finite element research on triangular patches did... |

68 |
The numerical evaluation of B-splines
- Cox
- 1972
(Show Context)
Citation Context ...ation of B-spline curves is due to him and is now known as the de Boor algorithm [40]. It is based on a recursion for B-splines which was independently discovered by de Boor, L. Mansfield, and M. Cox =-=[37]-=-. It was this recursion that made B-splines a truly viable tool in CAGD. Before its discovery, B-splines were defined using a tedious divided difference approach which was numerically very unstable. F... |

61 |
Inserting new knots into B-spline curves
- Boehm
- 1980
(Show Context)
Citation Context ...-to-Bézier conversion was found by W.Boehm [19]. Several algorithms were soon developed that simplified the mathematical treatment of B-spline curves; these include Boehm’s knot insertion algorithm 6 =-=[20]-=-, the Oslo algorithm by E. Cohen, T. Lyche, and R. Riesenfeld [32], and the introduction of the blossoming principle by L. Ramshaw [110] and P. de Casteljau [45]. The generalization of B-spline curves... |

51 |
On the dimension of bivariate spline spaces of smoothness r and degree d = 3r + 1. Numerische Mathematik 57
- Alfeld, Schumaker
- 1990
(Show Context)
Citation Context ..., first conceived by an automotive researcher, found their way into approximation theory in the 1980’s. Spaces of piecewise polynomials over triangulations were studied by L. Schumaker and Alfeld see =-=[1,2]-=-. Coons-like triangular patches were studied in the US during the 1970’s and 1980’s. See Barnhill et al [4], Barnhill and Gregory [7], or Nielson [102]. 8. Subdivision Surfaces At the 1974 CAGD confer... |

45 |
Shepard Method of ‘Metric interpolation’ to Bivariate and Multivariate
- GORDON, WIXON
- 1974
(Show Context)
Citation Context ...D. A scattered data interpolant is a function which interpolates the given data values and gives reasonable estimates in between. One of the first scattered data interpolants is Shepard’s method, see =-=[74,5,128]-=-. It computes the function value at an arbitrary (evaluation) point as a linear combination of all given function values, the coefficients being related to the distance of the data sites to the evalua... |

42 |
Computer Display of Curved Surfaces
- Gouraud
- 1971
(Show Context)
Citation Context ...herland’s 1963 development of interactive graphics that one could interactively change control polygons of Bézier or B-spline objects. Display techniques for parametric surfaces go back to H. Gouraud =-=[75]-=- and were later improved by B. Phong [105] and J. Blinn [18], all at the University of Utah. A fundamental display technique is ray tracing, due to T. Whitted [141]. It makes extensive use of computat... |

42 |
Transforming triangulations
- Lawson
- 1972
(Show Context)
Citation Context ...ional Geometry is that of triangulation algorithms. These are concerned with finding a set of triangles having a given 2D point set as vertices. The first algorithm was published in 1971 by C. Lawson =-=[94]-=-. An algorithmic connection between triangulations and Voronoi diagrams was presented by Green and Sibson [76]. The n−dimensional case was covered by D. Watson [140]. 2D triangulations are an importan... |

37 |
On the numerical condition of polynomials in Bernstein form
- Farouki, Rajan
- 1987
(Show Context)
Citation Context ... Some later results include conditions for C r joins between Bézier curves, 7s8 see Stärk [131], the discovery that Bézier curves are numerically more stable than other curve forms, see Farouki/Rajan =-=[62]-=-, and the development of the blossoming principle, see Ramshaw [110] and de Casteljau [46]. A symbolic technique for Bézier curves and surfaces was developed by M. Hosaka and F. Kimura [83], although ... |

36 |
Computer Display of Curved Surfaces
- Blinn
- 1978
(Show Context)
Citation Context ...could interactively change control polygons of Bézier or B-spline objects. Display techniques for parametric surfaces go back to H. Gouraud [75] and were later improved by B. Phong [105] and J. Blinn =-=[18]-=-, all at the University of Utah. A fundamental display technique is ray tracing, due to T. Whitted [141]. It makes extensive use of computations of the intersection between a ray and the scene to be d... |

36 | Interactive interpolation and approximation by Bézier polynomials
- Forrest
- 1972
(Show Context)
Citation Context ... Forrest. He realized that Bézier curves could be expressed in terms of Bernstein polynomials – i.e., in the form that de Casteljau had used since the late fifties! Forrest’s article on Bézier curves =-=[66]-=- was very influential and helped popularize Bézier curves considerably. The Rénault CAD/CAM system UNISURF was based entirely on Bézier curves and surfaces. It influenced developments by the French ai... |

34 |
F.: Design of Solids with Free-Form Surfaces
- KIMURA
- 1983
(Show Context)
Citation Context ...ubic boundary curves and cubic derivative information, this interpolant yields a rational patch. A “translation” of this approach into a Bézier-like form was carried out by H. Chiyokura and F. Kimura =-=[29,28]-=-. It led to the Japanese CAD/CAM system DESIGNBASE. Rectangular surfaces are a map of a rectangular domain into 3D. As a special case, we may map the domain to a 2D parametric surface, resulting in a ... |

30 |
Computational Geometry for Design and
- Faux, Pratt
- 1979
(Show Context)
Citation Context ... be regarded as the founding event of the field. It resulted in the widely influential proceedings [8]. The first textbook, “Computational Geometry for Design and Manufacture” by I. Faux and M. Pratt =-=[63]-=-, appeared in 1979. The journal “Computer Aided Geometric Design” was founded in 1984 by R. Barnhill and W. Boehm. Its cover is shown in Figure 1. Another early conference was one held in Paris in 197... |

29 | Splines as linear combinations of B-splines. A survey, in Approximation Theory
- Boor
- 1976
(Show Context)
Citation Context ... been around for a long time also. The first uses of these surfaces goes back to Finite Elements, 6A more general form of his algorithm was already known to de Boor from his work on B-splines in 1976 =-=[41]-=-. De Boor did not, however, realize the practical importance of this special case. 7As it turned out, this was not entirely true: some of Liming’s more complicated surface constructions do not allow a... |

29 |
Multivariable Curve Interpolation
- Ferguson
- 1964
(Show Context)
Citation Context ...cLaren developed a different kind of curve for the design of wings. They had the idea to piece cubic space curves together so that they formed composite curves which were overall twice differentiable =-=[97,64]-=-. These curves could easily interpolate to a set of points. They were referred to as spline curves since they minimize a functional similar to the physical properties of mechanical splines. The meanin... |

29 |
Smooth Interpolation without Twist Constraints. Computer Aided Geometric Design
- GREGORY
- 1974
(Show Context)
Citation Context ...on to the boundary curves. As a consequence, certain incompatible situations could arise. J. Gregory was the first to address this problem and also to devise a “compatibly corrected” interpolant, see =-=[77]-=-. When applied to cubic boundary curves and cubic derivative information, this interpolant yields a rational patch. A “translation” of this approach into a Bézier-like form was carried out by H. Chiyo... |

28 | Error bounds for spline interpolation.
- BIRKHOFF, BOOR
- 1964
(Show Context)
Citation Context ...for driving a milling tool. Parametric surfaces are well-suited for both tasks. The most popular of all surface methods was to become the tensor product surface. It was first introduced by C. de Boor =-=[39]-=- for the case of bicubic spline interpolation. Theoretical studies of parametric surfaces for the purpose of interpolation and approximation go back to [33,88,87,135,122,132] but had little influence ... |

27 |
Some piecewise polynomial alternatives to splines under tension
- Nielson
- 1974
(Show Context)
Citation Context ...called geometrically continuous or G 2 .Earlywork on this subject is due to G. Geise [70]. Interpolation schemes based on G 2 continuity were independently developed by J. Manning [98] and G. Nielson =-=[101]-=- in 1974. Later work on the subject includes B. Barsky’s β-splines [9], W. Boehm’s γ-splines [22], and H. Hagen’s τ-splines [79]. Shape does not only play a vital role in interpolation, but also in ap... |

27 |
Illumination for Computer-Generated Images
- PHONG
- 1973
(Show Context)
Citation Context ... graphics that one could interactively change control polygons of Bézier or B-spline objects. Display techniques for parametric surfaces go back to H. Gouraud [75] and were later improved by B. Phong =-=[105]-=- and J. Blinn [18], all at the University of Utah. A fundamental display technique is ray tracing, due to T. Whitted [141]. It makes extensive use of computations of the intersection between a ray and... |

24 |
The Beta-spline: A Local Representation Based on Shape Parameters and Fundamental Geometric Measures
- Barsky
- 1981
(Show Context)
Citation Context ... to G. Geise [70]. Interpolation schemes based on G 2 continuity were independently developed by J. Manning [98] and G. Nielson [101] in 1974. Later work on the subject includes B. Barsky’s β-splines =-=[9]-=-, W. Boehm’s γ-splines [22], and H. Hagen’s τ-splines [79]. Shape does not only play a vital role in interpolation, but also in approximation. The least squares method is the most widespread approxima... |

24 |
Spline-Blended Surface Interpolation Through Curve Networks
- Gordon
- 1969
(Show Context)
Citation Context .... These surfaces were used in the sixties by Ford (Coons was a consultant). A generalization, capable of interpolating a rectangular network of curves, was devised by W. Gordon at General Motors, see =-=[72, 71]-=-. All these methods are sometimes labeled ”transfinite interpolation,” in that they 3Private communication 4Clearly, he was unaware of de Boor’s paper [39] – it appeared in a journal not likely to be ... |

22 | Smooth interpolation in triangles
- Barnhill, Birkhoff, et al.
- 1973
(Show Context)
Citation Context ...s of piecewise polynomials over triangulations were studied by L. Schumaker and Alfeld see [1,2]. Coons-like triangular patches were studied in the US during the 1970’s and 1980’s. See Barnhill et al =-=[4]-=-, Barnhill and Gregory [7], or Nielson [102]. 8. Subdivision Surfaces At the 1974 CAGD conference at the University of Utah, one of the presenters was the graphics artist G. Chaikin. He presented a cu... |

22 |
Correction of local surface irregularities using reflection lines
- Klass
- 1980
(Show Context)
Citation Context ...owroom where the ceiling is lined with florescent light strips. Their reflections are carefully examined before the prototype is accepted. Early computer simulations of this procedure are reported in =-=[90]-=-. 11. Influences and Applications CAGD emerged through the influence of several areas in the 1950s and 1960s, but interactions with other fields of science and engineering were not limited to those ye... |

22 |
The side-vertex method for interpolation in triangles
- Nielson
- 1979
(Show Context)
Citation Context ...ions were studied by L. Schumaker and Alfeld see [1,2]. Coons-like triangular patches were studied in the US during the 1970’s and 1980’s. See Barnhill et al [4], Barnhill and Gregory [7], or Nielson =-=[102]-=-. 8. Subdivision Surfaces At the 1974 CAGD conference at the University of Utah, one of the presenters was the graphics artist G. Chaikin. He presented a curve generation method that did not fit the m... |

20 | An offset spline approximation for plane cubic splines - Klass - 1983 |

19 |
Fairing cubic B-spline curves, Computer Aided Geometric Design 4
- Farin, Rein, et al.
- 1987
(Show Context)
Citation Context ...ect, apply a fairing procedure. Such methods typically aim at removing noise from either data points or control polygons; early work is reported by J. Kjellander [89], J. Hoschek [86] and Farin et al =-=[60]-=-. A curve which is curvature continuous may not be twice differentiable. This fact, when properly exploited, leads to curve generation schemes which have more degrees of freedom than do “standard” spl... |

18 |
Dual Bézier curves and surfaces
- Hoschek
- 1983
(Show Context)
Citation Context ...atment goes back to Forrest’s PhD thesis [65]. A rational generalization of the de Casteljau algorithm was fiven by G. Farin [57] in 1983; a (dual) projective formulation was discovered by J. Hoschek =-=[84]-=-. 7. Triangular Patches There are (at least) two ways to describe a bivariate polynomial surface. One is as a tensor product, using a rectangular domain. The other one is to write it in terms of baryc... |

17 |
Algorithms for rational Bézier curves
- Farin
- 1983
(Show Context)
Citation Context ...cialized are conic sections, or rational quadratic Bézier curves. Their treatment goes back to Forrest’s PhD thesis [65]. A rational generalization of the de Casteljau algorithm was fiven by G. Farin =-=[57]-=- in 1983; a (dual) projective formulation was discovered by J. Hoschek [84]. 7. Triangular Patches There are (at least) two ways to describe a bivariate polynomial surface. One is as a tensor product,... |

16 |
Surface for computer aided design
- Coons
- 1964
(Show Context)
Citation Context ...icubic patch. 5 Ferguson’s bicubic patches were also known as F-patches, and were also attributed to S. Coons. Coons devised a simple formula to fit a patch between any four arbitrary boundary curves =-=[35]-=-, known as the bilinearly blended Coons patch. These surfaces were used in the sixties by Ford (Coons was a consultant). A generalization, capable of interpolating a rectangular network of curves, was... |

16 |
Riesenfeld, "B-spline Curves and Surfaces
- Gordon, F
- 1974
(Show Context)
Citation Context ...scussion, see [42]. Spline functions are important in approximation theory, but in CAGD, parametric spline curves are much more important. These were introduced in 1974 by R. Riesenfeld and W. Gordon =-=[73]-=- (the paper is a synopsis of Riesenfeld’s PhD thesis [112]) who realized 9s10 that de Boor’s recursive B-spline evaluation was the natural generalization of the de Casteljau algorithm. B-spline curves... |

15 | Procede de Definition Numerique des Courbes et Surfaces Non Mathematiques; Systeme UNISURF, Automatisme 13 - BEZIER - 1968 |

14 |
A matrix approach to the analysis of recursively generated B-spline surfaces
- Ball, Storry
- 1986
(Show Context)
Citation Context ...ill have singular points. These singular points hampered the analysis and practical use of subdivision surfaces. A first systematic investigation was carried out by D. Storry and A. Ball in 1986, see =-=[3]-=-, although there is an eigenvalue analysis of the Doo-Sabin process in the original paper [50]. Subsequently, subdivision surfaces gained more popularity, most notably in the computer animation indust... |

14 | Mathematical and practical possibilities of unisurf - Bezier - 1974 |

14 |
Biarc curves
- Bolton
- 1975
(Show Context)
Citation Context ...o a dense polygon as is needed for standard parametric splines. A drawback of circle splines is their piecewise constant and hence discontinuous curvature. The first developments are due to K. Bolton =-=[23]-=-, followed by M. Sabin [121]; a generalization to 3D was given by T. Sharrock [127]. 5. Rectangular Surfaces Parametric surfaces were well understood after early work by Gauss and Euler. They were imm... |

14 |
On the rendering of surfaces
- FORREST
- 1979
(Show Context)
Citation Context ...einsch, see [124,111], and also Powell [107]. For curves, curvature is a reliable shape measure. For surfaces, several such measures exist, including Gaussian, mean, or total curvatures. A.R. Forrest =-=[67]-=- was the first to use computer graphics for the interrogation of surface shape using curvatures as texture maps; see Fig. 7. Another important shape measure for surfaces comes from the automotive indu... |

14 | uOffset curves in the plane - Hoschek - 1985 |

14 |
Smoothing of cubic parametric splines
- Kjellander
- 1983
(Show Context)
Citation Context ...and inspect its shape: if it is imperfect, apply a fairing procedure. Such methods typically aim at removing noise from either data points or control polygons; early work is reported by J. Kjellander =-=[89]-=-, J. Hoschek [86] and Farin et al [60]. A curve which is curvature continuous may not be twice differentiable. This fact, when properly exploited, leads to curve generation schemes which have more deg... |

13 |
Cutting corners always works
- Boor
- 1987
(Show Context)
Citation Context ...d an iterative way to generate uniform quadratic B-spline curves, see [113]. In 1987, C. de Boor discovered that “corner cutting” generalizations of Chaikin’s algorithm also produce continuous curves =-=[43]-=-. He also pointed out that Chaikin’s algorithm was a special case of a class of algorithms described by G. de Rham much earlier [47,48]. Similar results were discovered by J. Gregory and R. Qu in 1988... |

13 |
On bounded polynomials in several variables
- Kellogg
- 1928
(Show Context)
Citation Context ...face. It was first introduced by C. de Boor [39] for the case of bicubic spline interpolation. Theoretical studies of parametric surfaces for the purpose of interpolation and approximation go back to =-=[33,88,87,135,122,132]-=- but had little influence on the development of industrial methods. In the late 1950s, parametric surfaces were studied at several companies in Europe and the U.S. The first published result is due to... |

12 |
Curvature Continuous Curves and Surfaces
- Boehm
- 1985
(Show Context)
Citation Context ...lation schemes based on G 2 continuity were independently developed by J. Manning [98] and G. Nielson [101] in 1974. Later work on the subject includes B. Barsky’s β-splines [9], W. Boehm’s γ-splines =-=[22]-=-, and H. Hagen’s τ-splines [79]. Shape does not only play a vital role in interpolation, but also in approximation. The least squares method is the most widespread approximation scheme. It was used ea... |

12 |
Finding intersections of B-spline represented geometries using recursive subdivision techniques. Computer Aided Geometric Design
- Dokken
- 1985
(Show Context)
Citation Context ...hoice (and tradition) to display/plot objects. Algorithms for this task were developed by many companies, and were mostly kept confidential. Some early published work is by W. Carlson [25], T. Dokken =-=[49]-=- Barnhill et al [6]. The earliest reference seems to be by M. Sabin [117]. Another important type of numerical algorithms are offset curves and surfaces – early work includes papers by R. Farouki [61]... |

12 | Exact offset procedures for simple solids - Farouki |

12 |
Practical Analytic Geometry with Application to Aircraft," The MacMillian Co
- Liming
- 1944
(Show Context)
Citation Context ...g marine and CAGD techniques may be found in [10,100,115,136]. Another key event originated in aeronautics. In 1944, R. Liming wrote a book entitled “Analytical Geometry with Application to Aircraft” =-=[95]-=-. Liming worked for the NAA (North American Aviation) during World War II; this company built fighter planes such as the legendary Mustang. In his book, classical drafting methods were combined with c... |

11 |
Définition numérique des courbes et surfaces II. Automatisme XII
- Bézier
- 1967
(Show Context)
Citation Context ...ow to detect shape imperfections in a direct way. Consequently, computer methods had to be developed that allowed easy assessment of shape. Among the first published methods are curve hodographs, see =-=[11]-=-. These are the plot of a curve’s first derivative curve. Since differentiation is a roughing process, curve imperfections appear “magnified” in the hodograph. Hodographs do not display the geometry o... |

11 |
Solid Modelling with DESIGNBASE
- Chiyokura
- 1988
(Show Context)
Citation Context ...ubic boundary curves and cubic derivative information, this interpolant yields a rational patch. A “translation” of this approach into a Bézier-like form was carried out by H. Chiyokura and F. Kimura =-=[29,28]-=-. It led to the Japanese CAD/CAM system DESIGNBASE. Rectangular surfaces are a map of a rectangular domain into 3D. As a special case, we may map the domain to a 2D parametric surface, resulting in a ... |

11 | Non-uniform corner cutting
- Gregory, Qu
- 1996
(Show Context)
Citation Context ...pecial case of a class of algorithms described by G. de Rham much earlier [47,48]. Similar results were discovered by J. Gregory and R. Qu in 1988 – although that work was only published in 1996, see =-=[78]-=-. Chaikin’s algorithm was the starting point for the initial work on subdivision surfaces, going back to two articles in the no. 6 volume of the journal Computer Aided Design, 1978. They were authored... |

10 |
Polynomial Interpolation to Boundary Data on Triangles
- Barnhill, Gregory
- 1975
(Show Context)
Citation Context ... over triangulations were studied by L. Schumaker and Alfeld see [1,2]. Coons-like triangular patches were studied in the US during the 1970’s and 1980’s. See Barnhill et al [4], Barnhill and Gregory =-=[7]-=-, or Nielson [102]. 8. Subdivision Surfaces At the 1974 CAGD conference at the University of Utah, one of the presenters was the graphics artist G. Chaikin. He presented a curve generation method that... |

9 |
Continuity conditions for spline curves
- Manning
- 1974
(Show Context)
Citation Context ...esulting curves are called geometrically continuous or G 2 .Earlywork on this subject is due to G. Geise [70]. Interpolation schemes based on G 2 continuity were independently developed by J. Manning =-=[98]-=- and G. Nielson [101] in 1974. Later work on the subject includes B. Barsky’s β-splines [9], W. Boehm’s γ-splines [22], and H. Hagen’s τ-splines [79]. Shape does not only play a vital role in interpol... |

8 |
Triangulation of scattered data in 3D space
- Choi, Chin, et al.
- 1988
(Show Context)
Citation Context ... 2D triangulations are an important preprocessing tool for many surface fitting operations. 3D point sets arise when physical objects are digitized. Methods to triangulate them go back to Choi et al. =-=[30]-=-. Computer Graphics is an area with many CAGD interactions. Computer Graphics needs CAGD to model objects to be displayed, and for the very same reason, CAGD needs Computer Graphics. It was only after... |

8 |
Un peu de mathématiques á propos d’une courbe plane. Elemente der
- Rham
- 1947
(Show Context)
Citation Context ...neralizations of Chaikin’s algorithm also produce continuous curves [43]. He also pointed out that Chaikin’s algorithm was a special case of a class of algorithms described by G. de Rham much earlier =-=[47,48]-=-. Similar results were discovered by J. Gregory and R. Qu in 1988 – although that work was only published in 1996, see [78]. Chaikin’s algorithm was the starting point for the initial work on subdivis... |

8 |
Cubic spline, a curve fitting routine
- Fowler, Wilson
- 1962
(Show Context)
Citation Context ...it was known to W. Boehm in 1972. 3 As an early alternative to parametric curves were explicit curve segments with individual local coordinate systems. These curves are known as Wilson-Fowler splines =-=[68]-=-. A similar curve type was employed in the TABCYL (TABulated CYLinder) routines of the APT (Automatic Programmed Tool) language. After the advent of parametric curves, these piecewise explicit curves ... |

8 |
Curvature profiles for plane curves
- Nutbourne, McLellan, et al.
- 1972
(Show Context)
Citation Context ...the curvature of a curve. Since it involves second order derivatives as well as first order ones, it is also a more sensitive tool. An early paper on the use of curvature plots is by Nutbourne et al. =-=[104]-=-. In unpublished work, curvature plots were used by H. Burchardt at General Motors. After de Boor left the company in 1964, interpolating cubic and quintic spline curves had become a tool of choice. S... |

7 |
An algorithm and data structure for 3D object synthesis using surface patch intersections
- CARLSON
- 1982
(Show Context)
Citation Context ... the method of choice (and tradition) to display/plot objects. Algorithms for this task were developed by many companies, and were mostly kept confidential. Some early published work is by W. Carlson =-=[25]-=-, T. Dokken [49] Barnhill et al [6]. The earliest reference seems to be by M. Sabin [117]. Another important type of numerical algorithms are offset curves and surfaces – early work includes papers by... |

7 |
une courbe plane
- Rham, Sur
- 1956
(Show Context)
Citation Context ...neralizations of Chaikin’s algorithm also produce continuous curves [43]. He also pointed out that Chaikin’s algorithm was a special case of a class of algorithms described by G. de Rham much earlier =-=[47,48]-=-. Similar results were discovered by J. Gregory and R. Qu in 1988 – although that work was only published in 1996, see [78]. Chaikin’s algorithm was the starting point for the initial work on subdivis... |

7 | Bézier polynomials over triangles and the construction of piecewise C r polynomials, TR/91
- Farin
- 1981
(Show Context)
Citation Context ...n gave conditions for smooth joins between adjacent triangular patches; G. Farin [55] gave conditions for C r joins. Early research on Bézier triangles focussed on equilateral domain triangles; Farin =-=[56,58]-=- discussed the case of arbitrarily shaped domain triangles. Bézier triangles, first conceived by an automotive researcher, found their way into approximation theory in the 1980’s. Spaces of piecewise ... |

7 |
Curves and Surfaces for Computer-Aided Design
- FORREST
- 1968
(Show Context)
Citation Context ...s work on curves and surfaces after being exposed to S. Coons’ ideas. His PhD thesis (Cambridge) includes work on shape classification of cubics, rational cubics, and generalizations of Coons patches =-=[65]-=-. M. Sabin worked for British Aircraft Corporation and was instrumental in developing the CAD system “Numerical Master Geometry.” He developed many algorithms that were later “reinvented.” This includ... |

6 |
A Case Study of Multivariate Piecewise Polynomials
- Alfeld
- 1987
(Show Context)
Citation Context ..., first conceived by an automotive researcher, found their way into approximation theory in the 1980’s. Spaces of piecewise polynomials over triangulations were studied by L. Schumaker and Alfeld see =-=[1,2]-=-. Coons-like triangular patches were studied in the US during the 1970’s and 1980’s. See Barnhill et al [4], Barnhill and Gregory [7], or Nielson [102]. 8. Subdivision Surfaces At the 1974 CAGD confer... |

6 |
Subdividing multivariate splines
- Boehm
- 1983
(Show Context)
Citation Context ...surfaces to a new G 1 subdivision surface type, see [96]. Its input control polygon can be any triangular mesh. Loop’s algorithm is based on a subdivision scheme for so-called box-splines by W. Boehm =-=[21]-=- and H. Prautzsch [108]. All three of the above algorithms produce C 1 (for Doo/Sabin and Loop) or C 2 (for Catmull/Clark and Loop) surfaces if the control meshes are regular, i.e., rectilinear or reg... |

6 |
Sapidis. Approximation with aesthetic constraints
- Burchard, Ayers, et al.
- 1994
(Show Context)
Citation Context ...hey are also curvature continuous. In many cases, however, this does not guarantee pleasant shape, and so Burchardt developed a proprietary scheme that was shape optimizing. A published version is in =-=[24]-=-. Shape optimization became an important element early on in the development of CAGD. Since interpolating spline curves were known to exhibit unwanted undulations, alternatives were studied, typically... |

6 |
Geometric spline curves. Computer Aided Geometric Design
- Hagen
- 1985
(Show Context)
Citation Context ...tinuity were independently developed by J. Manning [98] and G. Nielson [101] in 1974. Later work on the subject includes B. Barsky’s β-splines [9], W. Boehm’s γ-splines [22], and H. Hagen’s τ-splines =-=[79]-=-. Shape does not only play a vital role in interpolation, but also in approximation. The least squares method is the most widespread approximation scheme. It was used early on in most industries; publ... |

6 |
Curve fitting by splines in one variable
- POWELL
- 1970
(Show Context)
Citation Context ...typically the result of minimizing certain functionals (such as minimizing “wiggles”); an influential early example is the “smoothing spline” of Schoenberg and Reinsch, see [124,111], and also Powell =-=[107]-=-. For curves, curvature is a reliable shape measure. For surfaces, several such measures exist, including Gaussian, mean, or total curvatures. A.R. Forrest [67] was the first to use computer graphics ... |

4 |
Casteljau, Formes àpôles
- de
- 1985
(Show Context)
Citation Context ...o a cube and then deforming it using trivariate “Bézier cubes,” see [12,16] and Section 5. De Casteljau retired from Citroën in 1989 and became active in publishing. In 1985, he wrote “Formes áPôles,”=-=[45]-=- which introduced the concept of blossoming. 2 P. Bézier died in Paris in 1999. 4. Parametric Curves Curves were employed by draftsmen for centuries; the majority of these curves were circles, but som... |

4 |
Interpolationsformeln für Funktionene mehrerer Argumente, Skandinavisk Aktuarietidskrift 9
- Neder
- 1926
(Show Context)
Citation Context ...face. It was first introduced by C. de Boor [39] for the case of bicubic spline interpolation. Theoretical studies of parametric surfaces for the purpose of interpolation and approximation go back to =-=[33,88,87,135,122,132]-=- but had little influence on the development of industrial methods. In the late 1950s, parametric surfaces were studied at several companies in Europe and the U.S. The first published result is due to... |

4 |
Formulas for fitting a spline curve through a set of points
- MacLaren
- 1958
(Show Context)
Citation Context ...cLaren developed a different kind of curve for the design of wings. They had the idea to piece cubic space curves together so that they formed composite curves which were overall twice differentiable =-=[97,64]-=-. These curves could easily interpolate to a set of points. They were referred to as spline curves since they minimize a functional similar to the physical properties of mechanical splines. The meanin... |

3 |
1978] “General distortion of an ensemble of biparametric patches
- Bézier
- 1978
(Show Context)
Citation Context ...ree-dimensional Interactive Application). Bézier also invented a method to deform whole assemblies of surfaces by embedding them into a cube and then deforming it using trivariate “Bézier cubes,” see =-=[12,16]-=- and Section 5. De Casteljau retired from Citroën in 1989 and became active in publishing. In 1985, he wrote “Formes áPôles,”[45] which introduced the concept of blossoming. 2 P. Bézier died in Paris ... |

3 |
Cubic B-spline curves and surfaces in computer aided geometric design
- Boehm
- 1977
(Show Context)
Citation Context ...de Casteljau algorithm. B-spline curves include Bézier curves as a proper subset and soon became a core technique of almost all CAD systems. A first B-spline-to-Bézier conversion was found by W.Boehm =-=[19]-=-. Several algorithms were soon developed that simplified the mathematical treatment of B-spline curves; these include Boehm’s knot insertion algorithm 6 [20], the Oslo algorithm by E. Cohen, T. Lyche,... |

3 |
Rational bicubic surface patches
- Coons
- 1968
(Show Context)
Citation Context ...iform B-spline curves. The first systematic NURB treatment goes back to K. Versprille’s PhD thesis [139]. Versprille was a student of S. Coons’ who started working with rational curves in the sixties =-=[36]-=-. Coons’ work on rational curves also influenced A.R. Forrest, who wrote his PhD thesis in 1968 [65]. Versprille based rational curves and surfaces in homogeneous (or projective) space. This kind of g... |

2 |
Riesenfeld, editors. Computer Aided Geometric Design
- Barnhill, F
- 1974
(Show Context)
Citation Context ...niversity of Utah. That conference brought together researchers from the U.S. and from Europe and may be regarded as the founding event of the field. It resulted in the widely influential proceedings =-=[8]-=-. The first textbook, “Computational Geometry for Design and Manufacture” by I. Faux and M. Pratt [63], appeared in 1979. The journal “Computer Aided Geometric Design” was founded in 1984 by R. Barnhi... |

2 |
Smoothing of curves and surfaces. Computer Aided Geometric Design
- Hoschek
- 1985
(Show Context)
Citation Context ...hape: if it is imperfect, apply a fairing procedure. Such methods typically aim at removing noise from either data points or control polygons; early work is reported by J. Kjellander [89], J. Hoschek =-=[86]-=- and Farin et al [60]. A curve which is curvature continuous may not be twice differentiable. This fact, when properly exploited, leads to curve generation schemes which have more degrees of freedom t... |

2 |
The local dependence of least squares cubic splines
- Powell
- 1969
(Show Context)
Citation Context ...only play a vital role in interpolation, but also in approximation. The least squares method is the most widespread approximation scheme. It was used early on in most industries; publications include =-=[81,106,142]-=-. The least squares method lends itself to the inclusion of conditions that aim at the shape of the result, not just at the closeness of fit. These conditions are typically the result of minimizing ce... |

1 |
Properties of Shepard’s surfaces. Rocky Mtn
- Barnhill, Dube, et al.
- 1983
(Show Context)
Citation Context ...D. A scattered data interpolant is a function which interpolates the given data values and gives reasonable estimates in between. One of the first scattered data interpolants is Shepard’s method, see =-=[74,5,128]-=-. It computes the function value at an arbitrary (evaluation) point as a linear combination of all given function values, the coefficients being related to the distance of the data sites to the evalua... |

1 |
Surface / surface intersection. Computer Aided Geometric Design
- Barnhill, Farin, et al.
- 1987
(Show Context)
Citation Context ...) to display/plot objects. Algorithms for this task were developed by many companies, and were mostly kept confidential. Some early published work is by W. Carlson [25], T. Dokken [49] Barnhill et al =-=[6]-=-. The earliest reference seems to be by M. Sabin [117]. Another important type of numerical algorithms are offset curves and surfaces – early work includes papers by R. Farouki [61], J. Hoschek [85], ... |

1 |
Approximation von Funktionen bei einer und bei mehreren unabhängigen Veränderlichen
- Collatz
- 1956
(Show Context)
Citation Context ...face. It was first introduced by C. de Boor [39] for the case of bicubic spline interpolation. Theoretical studies of parametric surfaces for the purpose of interpolation and approximation go back to =-=[33,88,87,135,122,132]-=- but had little influence on the development of industrial methods. In the late 1950s, parametric surfaces were studied at several companies in Europe and the U.S. The first published result is due to... |

1 |
Graphical and analytical methods as applied to aircraft design
- Coons
- 1947
(Show Context)
Citation Context ...y adopted by U.S. aircraft companies. Figure 3 shows one of Liming’s constructions. Another researcher was also involved in the transition of aircraft drawings to computations; this was S. Coons, see =-=[34]-=-. Coons later gained fame for his work at MIT. Another early influential development for CAGD was the advent of numerical control (NC) in the 1950s. Early computers were capable of generating numerica... |

1 |
Courbes et surfaces àpôles
- Casteljau
(Show Context)
Citation Context ...face definitions from the very beginning, together with what is now known as the de Casteljau algorithm.sFigure 4. I. Sutherland’s Sketchpad system. Figure 5 shows a part of his 1963 technical report =-=[44]-=-. The breakthrough insight was to use control polygons (courbes àpôles), a technique that was never used before. Instead of defining a curve (or surface) through points on it, a control polygon utiliz... |

1 |
du Monceau. Eléments de l’Architecture Navale ou Traité Pratiquede la Construction des Vaissaux
- Duhamel
(Show Context)
Citation Context ...became popular in England in the 1600s. The classical “spline,” a wooden beam which is used to draw smooth curves, was probably invented then. The earliest available mention of a “spline” seems to be =-=[51]-=- from 1752. This “shipbuilding connection,” described by H. Nowacki [103], was the earliest use of constructive geometry to define free-form shapes, see Figure 2. More modern developments linking mari... |

1 |
Subsplines ueber Dreiecken
- Farin
- 1979
(Show Context)
Citation Context ...loped concepts that were closely related to Bézier triangles: Stancu [133], Frederickson [69], Sabin [120,121]. M. Sabin gave conditions for smooth joins between adjacent triangular patches; G. Farin =-=[55]-=- gave conditions for C r joins. Early research on Bézier triangles focussed on equilateral domain triangles; Farin [56,58] discussed the case of arbitrarily shaped domain triangles. Bézier triangles, ... |

1 |
Triangular spline interpolation / generalized triangular splines
- Frederickson
- 1971
(Show Context)
Citation Context ...riangles use trivariate Bernstein polynomials which did exist in approximation theory, several researchers developed concepts that were closely related to Bézier triangles: Stancu [133], Frederickson =-=[69]-=-, Sabin [120,121]. M. Sabin gave conditions for smooth joins between adjacent triangular patches; G. Farin [55] gave conditions for C r joins. Early research on Bézier triangles focussed on equilatera... |

1 |
Über berührende Kegelschnitte ebener Kurven
- Geise
- 1962
(Show Context)
Citation Context ... as shape 13s14 Figure 7. Gaussian curvature. Figure courtesy A.R. Forrest parameters and the resulting curves are called geometrically continuous or G 2 .Earlywork on this subject is due to G. Geise =-=[70]-=-. Interpolation schemes based on G 2 continuity were independently developed by J. Manning [98] and G. Nielson [101] in 1974. Later work on the subject includes B. Barsky’s β-splines [9], W. Boehm’s γ... |

1 |
The least-squares fitting of cubic splines to general data sets
- Hayes, Holladay
- 1974
(Show Context)
Citation Context ...only play a vital role in interpolation, but also in approximation. The least squares method is the most widespread approximation scheme. It was used early on in most industries; publications include =-=[81,106,142]-=-. The least squares method lends itself to the inclusion of conditions that aim at the shape of the result, not just at the closeness of fit. These conditions are typically the result of minimizing ce... |

1 |
A theory and methods for three dimensional free-form shape construction
- Hosaka, Kimura
- 1980
(Show Context)
Citation Context ...ouki/Rajan [62], and the development of the blossoming principle, see Ramshaw [110] and de Casteljau [46]. A symbolic technique for Bézier curves and surfaces was developed by M. Hosaka and F. Kimura =-=[83]-=-, although it was known to W. Boehm in 1972. 3 As an early alternative to parametric curves were explicit curve segments with individual local coordinate systems. These curves are known as Wilson-Fowl... |

1 |
Technical, physical, and mathematical principles of the finite element method
- Kolar, Kratochvil, et al.
- 1971
(Show Context)
Citation Context ....” The simplest type is the linear element, which is simply a planar triangular facet. Its use goes back to the very beginning of Ritz-Galerkin methods. Higher order elements use C 1 quintic elements =-=[92]-=- or C 1 split triangle cubic elements. The latter one, devised by Clough and Tocher as a finite element [31], gained some popularity in the context of scattered data interpolation, see [58]. From a hi... |

1 |
Système d’aide àladéfinition et à l’usinage des surfâces de carosserie
- Krautter, Parizot
- 1971
(Show Context)
Citation Context ...ut had no impact on any applications. De Casteljau’s work was kept a secret by Citroën for a long time. The first public mention of the algorithm (although not including a mention of the inventor) is =-=[93]-=-. W. Boehm was the first to give de Casteljau recognition for his work in the research community. He found out about de Casteljau’s technical reports and coined the term ”de Casteljau algorithm” in th... |

1 |
A curve-fitting method based on a variational criterion for smoothness
- Mehlum
- 1964
(Show Context)
Citation Context ... an important element early on in the development of CAGD. Since interpolating spline curves were known to exhibit unwanted undulations, alternatives were studied, typically involving curvature – see =-=[99,100,82,125]-=-. A different approach is to take an existing curve or surface and inspect its shape: if it is imperfect, apply a fairing procedure. Such methods typically aim at removing noise from either data point... |

1 |
Example of an existing system in the shipbuilding industry: the AUTOKON system
- Mehlum, Sorenson
- 1971
(Show Context)
Citation Context ...on,” described by H. Nowacki [103], was the earliest use of constructive geometry to define free-form shapes, see Figure 2. More modern developments linking marine and CAGD techniques may be found in =-=[10,100,115,136]-=-. Another key event originated in aeronautics. In 1944, R. Liming wrote a book entitled “Analytical Geometry with Application to Aircraft” [95]. Liming worked for the NAA (North American Aviation) dur... |

1 |
Splines in shipbuilding
- Nowacki
- 2000
(Show Context)
Citation Context ... beam which is used to draw smooth curves, was probably invented then. The earliest available mention of a “spline” seems to be [51] from 1752. This “shipbuilding connection,” described by H. Nowacki =-=[103]-=-, was the earliest use of constructive geometry to define free-form shapes, see Figure 2. More modern developments linking marine and CAGD techniques may be found in [10,100,115,136]. Another key even... |