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8.4 Stationary points of curvature of implicit surfaces

Now we provide a method to evaluate the stationary points of a curvature function $ C(x,y,z)$ of implicit surfaces
$\displaystyle f(x,y,z)=0\;.$     (8.88)

We want to maximize or minimize the function $ C(x,y,z)$ subject to a constraint (8.88). Introducing the Lagrange multiplier $ \lambda$ and the auxiliary function [166]
$\displaystyle \phi(x,y,z) = C(x,y,z) + \lambda f(x,y,z)\;,$     (8.89)

the necessary conditions for the auxiliary function $ \phi(x,y,z)$ to attain a local maximum or minimum, when no constraints are imposed, are
$\displaystyle \phi_x=0,\;\;\; \phi_y=0,\;\;\;\phi_z=0\;.$     (8.90)

Equations (8.90) together with (8.88) form four equations with four unknowns $ x$ , $ y$ , $ z$ and $ \lambda$ . The curvature functions $ C(x,y,z)$ 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 $ f(x,y,z)$ is a polynomial.

As an illustrative example, we will examine the stationary points of the minimum principal curvature $ \kappa_{min}(x,y,z)$ of an ellipsoid (3.83) [249] where we assume $ a\leq b \leq c$ . The auxiliary function becomes

$\displaystyle \phi = \kappa_{min}(x,y,z) + \lambda\left(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}-1\right)\;,$     (8.91)

and hence the system of equations to obtain the stationary points of minimum principal curvature of an ellipsoid reduce to
$\displaystyle \phi_x(x,y,z) = \kappa_x(x,y,z) + \frac{2x\lambda}{a^2}=0\;,$     (8.92)
$\displaystyle \phi_y(x,y,z) = \kappa_y(x,y,z) + \frac{2y\lambda}{b^2}=0\;,$     (8.93)
$\displaystyle \phi_z(x,y,z) = \kappa_z(x,y,z) + \frac{2z\lambda}{c^2}=0\;,$     (8.94)
$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}-1=0\;,$     (8.95)

where for simplicity we have set $ \kappa=\kappa_{min}$ .

After some algebraic manipulation, we obtain

$\displaystyle x(g_1 + g_2t + 4\lambda a^4b^2c^2 f_2^2 \sigma t) = 0\;,$     (8.96)
$\displaystyle y(h_1 + h_2t + 4\lambda a^2b^4c^2 f_2^2 \sigma t) = 0\;,$     (8.97)
$\displaystyle z(p_1 + p_2t + 4\lambda a^2b^2c^4 f_2^2 \sigma t) = 0\;,$     (8.98)
$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}-1=0\;,$     (8.99)
$\displaystyle t^2 - g_3 = 0\;,$     (8.100)
$\displaystyle \sigma^2 - f_2 = 0\;,$     (8.101)

where the last two equations are added through the introduction of auxiliary variables $ t$ and $ \sigma$ to remove the square roots of polynomials that appear in the denominator and numerator of the expression in (3.83), and
$\displaystyle f_1 = x^2 + y^2 + z^2 - a^2 - b^2 - c^2\;,$     (8.102)
$\displaystyle f_2 = \frac{x^2}{a^4} + \frac{y^2}{b^4} + \frac{z^2}{c^4}\;,$     (8.103)
$\displaystyle g_1 = -2f_1f_2a^4 - 8a^2b^2c^2f_2 + 3f_1^2\;,$     (8.104)
$\displaystyle g_2 = 2a^4f_2- 3f_1\;,$     (8.105)
$\displaystyle g_3 = f_1^2 - 4a^2b^2c^2f_2\;,$     (8.106)
$\displaystyle h_1 = -2f_1f_2b^4 - 8a^2b^2c^2f_2 + 3f_1^2\;,$     (8.107)
$\displaystyle h_2 = 2b^4f_2- 3f_1\;,$     (8.108)
$\displaystyle p_1 = -2f_1f_2c^4 - 8a^2b^2c^2f_2 + 3f_1^2\;,$     (8.109)
$\displaystyle p_2 = 2c^4f_2- 3f_1\;.$     (8.110)

Now the system consists of six equations with six unknowns $ x$ , $ y$ , $ z$ , $ \lambda$ , $ t$ and $ \sigma$ . 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 $ \kappa_{min}$ equal to $ \frac{a}{c^2}$ at $ (\pm a, 0, 0)$ , a local minimum $ \frac{b}{c^2}$ at $ (0, \pm b, 0)$ and a global maximum $ \frac{c}{b^2}$ at $ (0,0,\pm c)$ .

next up previous contents index
Next: 8.5 Contouring constant curvature Up: 8. Curve and Surface Previous: 8.3 Stationary points of   Contents   Index
December 2009