Computer Aided Engineering Design

216 COMPUTER AIDED ENGINEERING DESIGN

P 00 = {0, 0, 3}; P 10 = {1, 1, 3};P 20 = {1, 2, 3};P 30 = {0, 3, 3};

P 01 = {0, –1, 2};P 11 = {2, 1, 2};P 21 = {2, 2, 2};P 31 = {1, 3, 2};

P 02 = {0, –1, 1};P 12 = {1, 1, 1};P 22 = {2, 2, 1};P 32 = {1, 3, 1};

P 03 = {0, 0, 0}; P 13 = {0, 0, 0};P 23 = {0, 0, 0};P 33 = {0, 0, 0}

Figure 7.9 A triangular Bézier patch (a) without control points and (b) with control points

7.1.5 Triangular Surface Patch

Usually, a surface is created or modeled as a set of triangular or rectangular patches with continuity
conditions satisfied across the boundaries of adjoining patches. A way to generate a triangular patch
is described in Example 7.5.
Another way to model a triangular patch is to use three parameters u,v and w such that they are
constrained to sum to 1. With three parameters and a constraint, the patch still remains bi-parametric.
The triangular patch is defined by a set of control points rijk arranged in a triangular manner (Figure
7.10). Each control point is three dimensional and the indices i,j,k are such that 0 ≤i,j,k≤n,

Figure 7.10 Schematic of the placement of data points for a triangular patch

Case I: n = 2

r 020

r 011 r 110

r 002 r 101 r 200

Case II: n = 4

r 040

r 031 r 130

r 022 r 121 r 220

r 013 r 112 r 211 r 310

r 004 r 103 r 202 r 301 r 400

3 2 1 0 3 2 1 0 0

0.5

1

–1

0

1

2

3

(^0) 0.5
1
1.5
2
3
0
1
2
(a)
(b)