Physically Based Animation - Computer Animation: Algorithms and Techniques - page 237

y

z

curve to be removed

x

FIGURE 7.32
Two catenary curves supporting the same point.

FIGURE 7.33
Triangulation of constrained points in grid coordinates.

( Figure 7.33 ) . The vertices of the grid that fall on the lines of triangulation are positioned in three-space
according to the catenary equations. To find the catenary equation between two vertices ( x 1 , y 1 ), ( x 2 , y 2 )
of length L , see Equation 7.87 . Each of these triangles is repeatedly subdivided by constructing a cat-
enary from one of the vertices to the midpoint of the opposite edge on the triangle. This is done for all
three vertices of the triangle. The highest of the three catenaries so formed is kept and the others are
discarded. This breaks the triangle into two new triangles ( Figure 7.34 ) . This continues until all interior
vertices have been positioned.

FIGURE 7.34
Subdividing triangles.