Termination criterion for subdivision of triangular Bézier patch

doi:10.1016/S0097-8493(01)00158-3

Computers & Graphics

Volume 26, Issue 1, February 2002, Pages 67-74

https://doi.org/10.1016/S0097-8493(01)00158-3 Get rights and content

Abstract

A theorem regarding the termination criterion for subdivision of triangular Bézier patch is presented with detailed proof. As the number of times for subdividing a triangular Bézier patch can be a determined prior, according to the given tolerance, the subdividing process can be conduced consecutively without checking for linearity after each subdivision. The theorem has potentials in various CAD/CAM applications.

Introduction

Triangular Bézier patch was first introduced to CAGD by de Casteljau in internal technical reports (de Casteljau, 1959–1963). Since then, a great deal of work has been reported in these fields. Sabin [1] studied the triangular Bézier patch independently and his representation scheme is well known.

Subdivision of curves and surfaces are important operations in CAD/CAM because they produce sequences of linear segments or planar facets approximating the curves and surfaces within a given tolerance. Lane and Riesenfeld [2] proposed a termination criterion for subdivision of the Bézier curve and Wang [3] addressed the convergence problem regarding piecewise linear approximation of rational Bézier curve. Goldman [4] studied subdivision algorithms for triangular Bézier patch in 1983 and Chang, Davis [5] gave a practical formula for calculating the control vertices of the triangular sub-patches in 1984.

One interesting problem for subdivision operations is how to determine the number of subdivision times of a triangular Bézier patch a prior based on a given tolerance. In this paper, we present a theorem (Theorem 3) to deduce such a termination criterion.

Section snippets

Preliminaries

Assume a barycentric coordinate system over a domain triangle T:=ΔT₁T₂T₃ such that T₁=(1,0,0), T₂=(0,1,0) and T₃=(0,0,1), a triangular Bézier patch $B^{n} (u,v,w)$ of degree n over domain triangle T can be defined as $B^{n} (u,v,w)= ∑ 0⩽i,j,k⩽n,i+j+k=n B_{ijk}^{n} (u,v,w) f_{ijk} (0⩽u,v,w⩽1,u+v+w=1),$ where $f_{ijk} (i+j+k=n)$ denotes the control vertices of the triangular Bézier patch whose number amounts to (n+1)(n+2)/2 and $B_{ijk}^{n} (u,v,w)=n!/i!j!k! u^{i} v^{j} w^{k}$ are the respective Bernstein basic functions.

Let $D (r,s,t)$ be the 3D

Estimating the deviation of a triangular Bézier patch from its base triangle

Theorem 3

Suppose that $B^{n} (u,v,w)$ be a triangular Bézier patch which is defined over domain triangle T(T=ΔT₁T₂T₃) and ΔP₁P₂P₃ be a triangle inside T with its edges parallel to that of ΔT₁T₂T₃ (see Fig. 1, Fig. 2 and the barycentric coordinates of its vertices be (u_i,v_i,w_i), i=1,2,3. Let $B^{n} (u^{*},v^{*},w^{*})$ be a sub-patch of $B^{n} (u,v,w)$ corresponding to ΔP₁P₂P₃, then the maximum normal distance of $B^{n} (u^{*},v^{*},w^{*})$ to its base triangle $D (r^{*},s^{*},t^{*})$ will be no larger than a given tolerance ε, i.e.,

d(B^{n} (u^{*},v^{*},w^{*}), D (r^{*},s^{*},t^{*}

Geometric interpretations of Theorem 3

Assume that a triangular Bézier patch is subdivided into r²(r⩾r₀)sub-patches, then we can use 3D triangular facets defined by the corner vertices of these sub-patches to approximate the original triangular Bézier patch within the given tolerance ε. Therefore, Theorem 3 can be applied to intersection calculation, FEM mesh generation and STL output for triangular Bézier patch-based CAD systems.

An example is shown in Fig. 3. In this figure, a quartic triangular Bézier patch approximating a $18$

Acknowledgements

The work is jointly supported by the National Advanced Technology Project of China (No. 863-511-942-018) and the Research Fund for the Doctoral Program of Higher Education (No. 98033532).

References (6)

R.N. Goldman
Subdivision algorithms for Bézier triangles
Computer Aided Design
(1983)
G. Chang et al.
The convexity of Bernstein polynomicals over triangles
Journal of Approximation Theory
(1984)
M.A. Sabin
The use of piecewise forms for the numerical representation of shape. Ph.D. thesis
(1976)

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

Cited by (6)

A separation of the boundary geometry from the boundary functions in PIES for 3D problems modeled by the Navier–Lamé equation
2018, Computers and Mathematics with Applications
Citation Excerpt :
The boundary is modeled, as shown in Fig. 14 c, by triangular Bézier patches of degree 4. The single triangular patch approximates 1/8 of the sphere with coordinates of control points given in [50]. The full description of the boundary presented in Fig. 14 b requires 16 patches declared by 132 control points.
In this paper, we present a modification of the Somigliana identity for the 3D Navier–Lamé equation in order to analytically include in its mathematical formalism the boundary represented by Coons and Bézier parametric surface patches. As a result, the equations called the parametric integral equation system (PIES) with integrated boundary shape are obtained. The PIES formulation is independent from the boundary shape representation and it is always, for any shape, defined in the parametric domain and not on the physical boundary as in the traditional boundary integral equations (BIE). This feature is also helpful during numerical solving of PIES, as from a formal point of view, a separation between the approximation of the boundary and the boundary functions is obtained. In this paper, the generalized Chebyshev series are used to approximate the boundary functions. Numerical examples demonstrate the effectiveness of the presented strategy for boundary representation and indicate the high accuracy of the obtained results.
Nonelement boundary representation with Bézier surface patches for 3D linear elasticity problems in parametric integral equation system (PIES) and its solving using Lagrange polynomials
2018, Numerical Methods for Partial Differential Equations
Triangular Bézier surface patches in modeling shape of boundary geometry for potential problems in 3D
2013, Engineering with Computers
Evaluation of Boolean operations between free-form solids using Extended Simplicial Chains and PN triangles
2011, Visual Computer
Triangular bézier patches in modelling smooth boundary surface in exterior helmholtz problems solved by the PIES
2009, Archives of Acoustics
Adaptive trimming of cubic triangular Bézier patches
2006, Ibero-American Symposium in Computer Graphics, SIACG 2006

View full text

Technical SectionTermination criterion for subdivision of triangular Bézier patch

Abstract

Introduction

Section snippets

Preliminaries

Estimating the deviation of a triangular Bézier patch from its base triangle

Geometric interpretations of Theorem 3

Acknowledgements

Computer Aided Design

Journal of Approximation Theory

The use of piecewise forms for the numerical representation of shape. Ph.D. thesis

Technical Section
Termination criterion for subdivision of triangular Bézier patch