8.2.1 Gaussian curvature

To formulate the governing equations for computing the stationary points of Gaussian curvature within the domain, we substitute (3.46) into (8.29) which yields [255]

where

Polynomial and its partial derivatives are given by

where

and

As , (8.35) are satisfied if

which are two simultaneous bivariate polynomial equations of degree , in and , respectively. For example, if the input surface is a bicubic Bézier patch, the degrees of the two simultaneous bivariate polynomial equations become (23, 24) and (24, 23) in and . System (8.45) can be solved robustly with the IPP algorithm described in Chap. 4 (see [255] for details).

The stationary points along the four boundary edges are easily obtained by solving the four univariate polynomial equations,

using the IPP algorithm described in Chap. 4 (see [255] for details).

Example 8.2.1. A hyperbolic paraboloid , (bilinear surface), illustrated in Fig. 3.4, can be expressed in a Bézier form as

where , , and . The first and second fundamental form coefficients are readily computed using (3.63) (3.64) (3.65)

Thus the Gaussian curvature is given by

The Gaussian curvature is always negative and hence all the points are hyperbolic.

The stationary points within the parameter domain are given by finding the roots of . Since

and from (8.40) through (8.43)

and from (8.38) and (8.39)

hence we have

Thus the stationary point within the domain is given by , which coincides with one of the corner points of the bilinear surface patch. Using Theorem 7.3.1, we find that is a minimum.

At the remaining of corner points of the patch, we can readily compute

Stationary points along parameter domain boundaries are obtained by solving the following four univariate equations

The roots are , , and , thus they coincide with the corner points , , and . Therefore, the range of the Gaussian curvature is .