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
(8.88)
We want to maximize
or minimize the function
subject to a constraint
(8.88). Introducing the Lagrange
multiplier
and the auxiliary function
[166]
(8.89)
the necessary conditions for the auxiliary function
to attain a
local maximum or minimum, when no constraints are imposed, are
(8.90)
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
(8.91)
and hence the system of equations to obtain the stationary points of minimum
principal curvature of an ellipsoid reduce to
(8.92)
(8.93)
(8.94)
(8.95)
where for simplicity we have set
.
After some algebraic manipulation, we obtain
(8.96)
(8.97)
(8.98)
(8.99)
(8.100)
(8.101)
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
(8.102)
(8.103)
(8.104)
(8.105)
(8.106)
(8.107)
(8.108)
(8.109)
(8.110)
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
.