8.4 Stationary points of curvature of implicit surfaces

Now we provide a method to evaluate the stationary points of a curvature function of implicit surfaces

We want to maximize or minimize the function subject to a constraint (8.88). Introducing the Lagrange multiplier and the auxiliary function [166]

the necessary conditions for the auxiliary function to attain a local maximum or minimum, when no constraints are imposed, are

Equations (8.90) together with (8.88) form four equations with four unknowns , , and . The curvature functions including Gaussian, mean and principal curvatures are evaluated using the procedure described in Sect. 3.5.2. In a manner similar to parametric surfaces, the denominator and the numerator of the curvature functions consist of polynomials and square root of polynomials if is a polynomial.

As an illustrative example, we will examine the stationary points of the minimum principal curvature of an ellipsoid (3.83) [249] where we assume . The auxiliary function becomes

and hence the system of equations to obtain the stationary points of minimum principal curvature of an ellipsoid reduce to

where for simplicity we have set .

After some algebraic manipulation, we obtain

where the last two equations are added through the introduction of auxiliary variables and to remove the square roots of polynomials that appear in the denominator and numerator of the expression in (3.83), and

Now the system consists of six equations with six unknowns , , , , and . Since the degree of the polynomials is low we can solve the system by a symbolic manipulation program such as MATHEMATICA [446], MAPLE [51], which gives a global minimum of equal to at , a local minimum at and a global maximum at .