Triangular patch modeling using combination method

doi:10.1016/S0167-8396(02)00148-6

Computer Aided Geometric Design

Volume 19, Issue 8, October 2002, Pages 645-662

https://doi.org/10.1016/S0167-8396(02)00148-6 Get rights and content

Abstract

A new method for constructing triangular patches is presented. A triangular patch that interpolates given boundary curves and cross-boundary slopes is formed by blending three traditional side-vertex interpolation operators (Nielson, 1979) with a new, interior interpolation operator. The new operator is the solution of an interpolation process that interpolates both the interior and the boundary of the triangular domain. The interior interpolation operator has better approximation precision on the interior of the triangle than the side-vertex operators. The constructed triangular patch reproduces polynomial surfaces of degree four. Comparison results of the new method with the side-vertex method (Nielson, 1979) are included.

References (17)

R.E. Barnhill et al.
Smooth interpolation in triangles
J. Approx. Theory
(1973)
P. Charrot et al.
A pentagonal surface patch for computer aided geometric design
Computer Aided Geometric Design
(1984)
T.A. Foley et al.
Hybrid cubic Bézier triangle patches
H. Hagen
Geometric surface patches without twist constraints
Computer Aided Geometric Design
(1986)
H. Hagen
Curvature continuous triangular interpolants
S. Kuriyama
Surface modeling with an irregular network of curves via sweeping and blending
Computer-Aided Design
(1994)
C.L. Lawson
Software for $C^{1}$ surface interpolation
G.M. Nielson
The side vertex method for interpolation in triangles
J. Approx. Theory
(1979)

There are more references available in the full text version of this article.

Cited by (19)

An extension of the Bézier model
2011, Applied Mathematics and Computation
Citation Excerpt :
Triangular surface modeling method attracts the attention of many researchers because of its great potentials of constructing complex shape. However, after the theory of triangular Bernstein–Bézier surface is ripe, only a few literatures did research on the extension of triangular surface to improve the classical triangular Bernstein–Bézier surface; see [22–28]. In these literatures, only the surfaces presented in [27,28] enjoy shape adjustable property.
In this paper, we first construct a new kind of basis functions by a recursive approach. Based on these basis functions, we define the Bézier-like curve and rectangular Bézier-like surface. Then we extend the new basis functions to the triangular domain, and define the Bernstein–Bézier-like surface over the triangular domain. The new curve and surfaces have most properties of the corresponding classical Bézier curve and surfaces. Moreover, the shape parameter can adjust the shape of the new curve and surfaces without changing the control points. Along with the increase of the shape parameter, the new curve and surfaces approach the control polygon or control net. In addition, the evaluation algorithm for the new curve and triangular surface are provided.
Constructing C1 triangular patch of degree six by the Boolean sum of an approximation operator and an interpolation operator
2011, Applied Mathematical Modelling
Citation Excerpt :
An approach for constructing parametric triangular surface with G1 continuity over a given triangular mesh is proposed in Ref. [15]. Zhang et al., developed a new method [16] to construct a curved triangular patch by combining four interpolation operators, one interior interpolation operator and three side-vertex operators [11]. In Ref. [17], a C1 condition between two cubic triangular patches was derived using mixed directional derivatives, and then an approach for constructing a C1 triangular patch around a corner was presented.
A new method to construct C¹ triangular patches which satisfy the given boundary curves and cross-boundary slopes is presented. The Boolean sum of an approximation operator and an interpolation operator is employed to construct the triangular patch. The approximation operator is used to construct a polynomial patch of degree six. The polynomial of degree six affords more freedoms, which makes the approximation operator not only approximate the given boundary interpolation conditions but also have a better approximation precision in the interior of the triangle, so that the triangular patch has a better precision on both the boundary and the interior of the triangular domain. The interpolation operator is utilized to build an interpolation patch which satisfies the given boundary conditions. The Boolean sum of the approximation and interpolation patches forms the triangular patch. Comparison results of the new method with other three methods are given.
Constructing parametric triangular patches with boundary conditions
2008, Progress in Natural Science
The problem of constructing a parametric triangular patch to smoothly connect three surface patches is studied. Usually, these surface patches are defined on different parameter spaces. Therefore, it is necessary to define interpolation conditions, with values from the given surface patches, on the boundary of the triangular patch that can ensure smooth transition between different parameter spaces. In this paper we present a new method to define boundary conditions. Boundary conditions defined by the new method have the same parameter space if the three given surface patches can be converted into the same form through affine transformation. Consequently, any of the classic methods for constructing functional triangular patches can be used directly to construct a parametric triangular patch to connect given surface patches with $G^{1}$ continuity. The resulting parametric triangular patch preserves precision of the applied classic method.
C<sup>1</sup> triangular Coons surface construction and image interpolation based on new side-side and side-vertex interpolation operators
2020, PLoS ONE
Bivariate Pólya basis and Pólya triangular patch
2014, Journal of Information and Computational Science
Visualization and surface rendering based on medical image
2012, Computer-Aided Design and Applications

View all citing articles on Scopus

¹: Supported by Ford and Honda. Currently with the School of Computer Science and Technology, Shandong University, China.

²: Supported by NSF(9722728), Ford and Honda.

View full text

Triangular patch modeling using combination method

Abstract

J. Approx. Theory

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer-Aided Design

J. Approx. Theory