Surfaces in computer aided geometric design: a survey with new results

doi:10.1016/0167-8396(85)90002-0

Computer Aided Geometric Design

Volume 2, Issues 1–3, September 1985, Pages 1-17

Presented at Oberwolfach 12 November 1984

https://doi.org/10.1016/0167-8396(85)90002-0 Get rights and content

Abstract

‘Surfaces in Computer Aided Geometric Design’ focuses on the representation and design of surfaces in a computer graphics environment. This new area has the dual attractions of interesting research problems and important applications. The subject can be approached from two points of view: The design of surfaces which includes the interactive modification of geometric information and the representation of surfaces for which the geometric information is relatively fixed. Design takes place in 3-space whereas representation can be higher dimensional. ‘Surfaces in CAGD’ can be traced from its inception in rectangular Coons patches and Bezier patches to triangular patches which are current research topics. Triangular patches can interpolate and approximate to arbitrarily located data and require the preprocessing steps of triangulation and derivative estimation. New contouring methods have been found using these triangular patches. Finally, multidimensional interpolation schemes have been based on tetrahedral interpolants and are illustrated by surfaces in 4-space by means of color computer graphics.

References (109)

P. Alfeld
A trivariate Clough-Tocher scheme for tetrahedral data
Computer Aided Geometric Design
(1984)
P. Alfeld
A bivariate C² Clough-Tocher scheme
Computer Aided Geometric Design
(1984)
R.E. Barnhill
Coons' patches
Computers in Industry
(1982)
R.E. Barnhill et al.
Smooth interpolation in triangles
J. Approx. Theory
(1973)
R.E. Barnhill et al.
A new twist for computer aided geometric design
Computer Graphics and Image Processing
(1978)
R.E. Barnhill et al.
Smooth interpolation over hypercubes
Computer Aided Geometric Design
(1984)
G. Chang et al.
The convexity of Bernstein polynomials over triangles
J. Approx. Theory
(1984)
G. Farin
A construction for the visual C¹ continuity of polynomial surface patches
Computer Graphics and Image Processing
(1982)
Farin
Visually C² cubic splines
Computer Aided Design
(1982)
J.A. Gregory et al.
A C¹ triangular interpolation patch for computer aided geometric design
Computer Graphics and Image Processing
(1980)

L. Mansfield

Interpolation to boundary data in tetrahedra with applications to compatible finite elements

J. Math. Anal. Appl.

(1976)

G.M. Nielson

The side-vertex method for interpolation in triangles

J. Approx. Theory

(1979)

C.S. Petersen

Adaptive contouring of three-dimensional surfaces

Computer Aided Geometric Design

(1984)

P. Alfeld

Multivariate scattered data derivative generation by functional minimization

P. Alfeld

A discrete C¹ interpolant for tetrahedral data

Rocky Mountain J. Math.

(1984)

P. Alfeld et al.

A transfinite C² interpolant over triangles

Rocky Mountain J. Math

(1984)

P.R. Arner

Hidden surface elimination

R.E. Barnhill

Representation and approximation of surfaces

R.E. Barnhill

Computer aided surface representation and design

R.E. Barnhill

A survey of the representation and design of surfaces

IEEE Computer Graphics Appl.

(1983)

R.E. Barnhill

A survey of transfinite patch methods

(1985)

(1983)

R.E. Barnhill et al.

Approximation

R.E. Barnhill et al.

Properties of Shepard's surfaces

Rocky Mountain J. Math

(1983)

R.E. Barnhill et al.

C¹ quintic interpolation over triangles: two explicit representations

Internat. J. Numer. Methods Engrg.

(1981)

R.E. Barnhill et al.

Polynomial interpolation to boundary data on triangles

Math. Computation

(1975)

R.E. Barnhill et al.

Sard kernel theorems on triangular domains with application to finite element error bounds

Numerische Math.

(1976)

R.E. Barnhill et al.

Interpolation remainder theory from Taylor expansions with nonrectangular domains of influence

Numerische Math.

(1976)

R.E. Barnhill et al.

Visual continuity for space curves and surfaces

R.E. Barnhill et al.

Three- and four-dimensional surfaces

Rocky Mountain J. Math.

(1984)

Surfaces, a special issue of the Rocky Mountain J. Math.

(1984)

R.E. Barnhill et al.

Surface representation for the graphical display of structured data

R.E. Barnhill et al.

A multidimensional surface problem: pressure on a wing

(1985)

(1974)

R.E. Barnhill et al.

Multistage trivariate surfaces

Rocky Mountain J. Math.

(1984)

R.E. Barnhill et al.

A geometric interpretation of convexity conditions for surfaces

Computer Aided Geometric Design

(1985)

B.A. Barsky et al.

Geometric continuity of parametric curves

P. Bézier

Définition numérique des courbes et surfaces

Automatisme

(1966)

P. Bézier

Définition numérique des courbes et surfaces (II)

Automatisme

(1967)

P. Bézier

Numerical Control, Mathematics and Application

(1972)

W. Boehm et al.

A survey of curve and surface methods in CAGD

Computer Aided Geometric Design

(1984)

C. de Boor

A Practical Guide to Splines

(1978)

A. Bowyer

Computing Dirichlet tessellations

Computer J.

(1981)

P. Brunet

Increasing the smoothness of bicubic spline surfaces

(1985)

P. de Casteljau

Outillage méthods calcul

(1959)

P. de Casteljau

Courbes et surfaces à pôles

(1963)

G. Chang et al.

An improved condition for the convexity of Bernstein-Bézier surfaces over triangles

Computer Aided Geometric Design

(1985)

R.W. Clough et al.

Finite element stiffness matrices for analysis of plates in bending

S.A. Coons

Surfaces for computer aided design

(1964)

(1967)

P.J. Davis

Interpolation and Approximation

(1975)

Cited by (82)

Lupaş type Bernstein operators on triangles based on quantum analogue
2021, Alexandria Engineering Journal
The purpose of the paper is to introduce new analogues of Lupaş type Bernstein operators $(B_{m, q}^{u} f) (u, v)$ and $(B_{n, q}^{v} f) (u, v)$ , their products $(P_{mn, q} f) (u, v)$ and $(Q_{nm, q} f) (u, v)$ and their Boolean sums $(S_{mn, q} f) (u, v)$ and $(T_{nm, q} f) (u, v)$ on triangle $T_{h}$ , which interpolate a given function on the some edges and at the vertices of triangle using quantum analogue. Based on Peano’s theorem and using modulus of continuity, the remainders of the approximation formula of corresponding operators are evaluated. It has been shown that parameter q will provide more flexibility for approximation.
h-Bernstein basis functions over a triangular domain
2020, Computer Aided Geometric Design
Citation Excerpt :
A fundamental tool in Computer Aided Geometric Design (CAGD) to handle curves and surfaces is the Bernstein-Bézier technique (see e.g. Barnhill, 1985; Farin, 1986; Lai and Schumaker, 2007; Prautzsch et al., 2002), which is a geometrically intuitive and meaningful technique, leading to constructive numerically robust algorithms.
In this paper, we introduce and study new h-Bernstein basis functions over a triangular domain. In particular, after defining the h-Bernstein polynomial functions of degree n, we prove their algebraic and geometric properties, such as partition of unity and degree elevation and we show that they form a basis for the space of polynomials of total degree less than or equal to n on a triangle. Then, we propose the h-de Casteljau algorithm and we prove the Marsden identity.
On numerical quadrature for C<sup>1</sup> quadratic Powell–Sabin 6-split macro-triangles
2019, Journal of Computational and Applied Mathematics
The quadrature rule of Hammer and Stroud (1956) for cubic polynomials has been shown to be exact for a larger space of functions, namely the $C^{1}$ cubic Clough–Tocher spline space over a macro-triangle if and only if the split-point is the barycentre of the macro-triangle Kosinka and Bartoň (2018). We continue the study of quadrature rules for spline spaces over macro-triangles, now focusing on the case of $C^{1}$ quadratic Powell–Sabin 6-split macro-triangles. We show that the 3-node Gaussian quadrature(s) for quadratics can be generalised to the $C^{1}$ quadratic Powell–Sabin 6-split spline space over a macro-triangle for a two-parameter family of inner split-points, not just the barycentre as in Kosinka and Bartoň (2018). The choice of the inner split-point uniquely determines the positions of the edge split-points such that the whole spline space is integrated exactly by a corresponding polynomial quadrature. Consequently, the number of quadrature points needed to exactly integrate this special spline space reduces from twelve to three.
For the inner split-point at the barycentre, we prove that the two 3-node quadratic polynomial quadratures of Hammer and Stroud exactly integrate also the $C^{1}$ quadratic Powell–Sabin spline space if and only if the edge split-points are at their respective edge midpoints. For other positions of the inner and edge split-points we provide numerical examples showing that three nodes suffice to integrate the space exactly, but a full classification and a closed-form solution in the generic case remain elusive.
Bernstein-type operators on a triangle with all curved sides
2011, Applied Mathematics and Computation
Citation Excerpt :
There were constructed operators of Lagrange, Hermite and Birkhoff type that interpolate a given function and certain of its derivatives on one, two or all sides of the triangle (see, e.g., [4–6,12–15,23,24,26]). In order to match all the boundary information on a curved domain (as in Dirichlet, Neumann or Robin boundary conditions for differential equation problems) there were considered interpolation operators on triangles with curved sides (one, two or all curved sides), many of them in connection with finite element method and computer aided geometric design (see, e.g., [2,3,7,8,16,17,20,24,25]). In some of these papers the triangle with curved sides is first transformed in a triangle with straight sides and then it is studied the corresponding interpolation problem (see, e.g., [24]).
Correcting non-linear lens distortion in cameras without using a model
2010, Optics and Laser Technology
Citation Excerpt :
According to [30,31], techniques for surface interpolation of scattered data include triangulation (or tetrahedrization)-based methods, inverse distance weighted methods, radial basis function methods, natural neighbour methods, as well as a data fitting method. Triangulation-based methods are local and, hence, capable of treating efficiently large data sets [32–36]. They are computationally simple (once the prerequisite task of triangulation or tetrahedrization has been accomplished) and relatively accurate.
Camera lens distortion calibration is the first step in resolving any metric application with a camera. To date, lens distortion was corrected using some existing lens distortion non-metric or self-calibration methods. Using a lens distortion model means defining a global rule to correct the entire image. This global rule does not take into account particular lens distortion effects not represented by the model. Moreover, to calibrate the model, only some features of the scene such as straight lines, circles or vanishing points are used. Since only the feature of the scene used to calibrate the model is guaranteed by the distortion rectification, it is certain that the model will not be precise. The result is an approximation of the real image distortion.
To improve the lens distortion rectification, a method without using a model is proposed. Using a set of control points distributed across the entire image, they are corrected to assure all the restrictions of the scene. With both sets of points, the points detected in the image and the undistorted ones, image local transformations are defined considering only nearby control points. Rather than calibrating a global model, local functions are characterized. The distortion correction is defined by a rectification surface composed of local surface patches each influenced by nearby control points. This method is more sensitive to local deformations and allows the image to be corrected in accordance with its distortion.
CONSTRUCTION OF POLYNOMIAL PRESERVING COCHAIN EXTENSIONS BY BLENDING
2023, Mathematics of Computation

View all citing articles on Scopus

View full text

Surfaces in computer aided geometric design: a survey with new results

Abstract

Computer Aided Geometric Design

Computer Aided Geometric Design

Computers in Industry

J. Approx. Theory

Computer Graphics and Image Processing

Computer Aided Geometric Design

J. Approx. Theory

Computer Graphics and Image Processing

Computer Aided Design

Computer Graphics and Image Processing

J. Math. Anal. Appl.

J. Approx. Theory

Computer Aided Geometric Design

Multivariate scattered data derivative generation by functional minimization

A discrete C1 interpolant for tetrahedral data

Rocky Mountain J. Math.

A transfinite C2 interpolant over triangles

Rocky Mountain J. Math

Hidden surface elimination

Representation and approximation of surfaces

Computer aided surface representation and design

A survey of the representation and design of surfaces

IEEE Computer Graphics Appl.

A survey of transfinite patch methods

Approximation

Properties of Shepard's surfaces

Rocky Mountain J. Math

C1 quintic interpolation over triangles: two explicit representations

Internat. J. Numer. Methods Engrg.

Polynomial interpolation to boundary data on triangles

Math. Computation

Sard kernel theorems on triangular domains with application to finite element error bounds

Numerische Math.

Interpolation remainder theory from Taylor expansions with nonrectangular domains of influence

Numerische Math.

Visual continuity for space curves and surfaces

Three- and four-dimensional surfaces

Rocky Mountain J. Math.

Surfaces, a special issue of the Rocky Mountain J. Math.

Surface representation for the graphical display of structured data

A multidimensional surface problem: pressure on a wing

Multistage trivariate surfaces

Rocky Mountain J. Math.

A geometric interpretation of convexity conditions for surfaces

Computer Aided Geometric Design

Geometric continuity of parametric curves

Définition numérique des courbes et surfaces

Automatisme

Définition numérique des courbes et surfaces (II)

Automatisme

Numerical Control, Mathematics and Application

A survey of curve and surface methods in CAGD

Computer Aided Geometric Design

A Practical Guide to Splines

Computing Dirichlet tessellations

Computer J.

Increasing the smoothness of bicubic spline surfaces

Outillage méthods calcul

Courbes et surfaces à pôles

An improved condition for the convexity of Bernstein-Bézier surfaces over triangles

Computer Aided Geometric Design

Finite element stiffness matrices for analysis of plates in bending

Surfaces for computer aided design

Interpolation and Approximation

A discrete C¹ interpolant for tetrahedral data

A transfinite C² interpolant over triangles

C¹ quintic interpolation over triangles: two explicit representations