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8.2.3 Principal curvatures

For obtaining the stationary points of principal curvature $ \kappa$ within the domain, the simultaneous bivariate equations (8.29) become [255]
$\displaystyle \kappa_u(u,v) = \frac{f_1(u,v) \pm f_2(u,v)\sqrt{f_3(u,v)}}{2S^5(u,v)\sqrt{f_3(u,v)}} = 0, \;\;\; 0 \leq u, v \leq 1\;,$     (8.60)
$\displaystyle \kappa_v(u,v) = \frac{g_1(u,v) \pm g_2(u,v)\sqrt{f_3(u,v)}}{2S^5(u,v)\sqrt{f_3(u,v)}} = 0, \;\;\; 0 \leq u, v \leq 1\;.$      

The plus and minus signs correspond to the maximum and minimum principal curvatures, and $ f_1(u, v)$ , $ f_2(u,v)$ , $ f_3(u,v)$ , $ g_1(u,v)$ and $ g_2(u,v)$ are polynomials of degree $ (14m-9, 14n-8)$ , $ (9m-6, 9n-5)$ , $ (10m-6, 10n-6)$ , $ (14m-8, 14n-9)$ , $ (9m-5, 9n-6)$ in $ u$ and $ v$ parameters and are given by
$\displaystyle f_1(u,v) = (BB_u - 2 A_uS^2)S^2 + (8AS^2 - 3B^2)({\bf S}\cdot{\bf S}_u)\;,$     (8.61)
$\displaystyle f_2(u,v) = B_uS^2 - 3({\bf S}\cdot{\bf S}_u)B\;,$     (8.62)
$\displaystyle f_3(u,v) = B^2 - 4AS^2\;,$     (8.63)
$\displaystyle g_1(u,v) = (BB_v - 2 A_vS^2)S^2 + (8AS^2 - 3B^2)({\bf S}\cdot{\bf S}_v)\;,$     (8.64)
$\displaystyle g_2(u,v) = B_vS^2 - 3({\bf S}\cdot{\bf S}_v)B\;.$     (8.65)

Assuming $ f_3 \neq 0$ and $ S \neq 0$ , we obtain
$\displaystyle f_1(u,v) \pm f_2(u,v)\sqrt{f_3(u,v)} = 0, \;\;\; 0 \leq u,v \leq 1\;,$     (8.66)
$\displaystyle g_1(u,v) \pm g_2(u,v)\sqrt{f_3(u,v)} = 0, \;\;\; 0 \leq u,v \leq 1\;.$      

These are two simultaneous bivariate irrational equations involving polynomials and square roots of polynomials which arise from the analytical expressions of principal curvatures. We can introduce an auxiliary variable $ \tau$ such that $ \tau^2=f_3$ to remove the radical and transform the problem into a system of three trivariate polynomial equations of degree $ (14m-9, 14n-8,1)$ , $ (14m-8, 14n-9,1)$ , $ (10m-6, 10n-6,2)$ in $ u$ , $ v$ and $ \tau$ . For a bicubic Bézier patch the degrees of the trivariate polynomial equations are (33,34,1), (34,33,1) and (24,24,2). The resulting system can be solved with the IPP algorithm of Chap. 4 (see [255] for details).

For the stationary points of principal curvatures along the boundary, we need to solve the following four univariate irrational equations involving polynomials and square roots of polynomials

$\displaystyle f_1(u,0) \pm f_2(u,0)\sqrt{f_3(u,0)} = 0,\;\; f_1(u,1) \pm f_2(u,1)\sqrt{f_3(u,1)} = 0,\;\; 0 \leq u \leq 1\;,$     (8.67)
$\displaystyle g_1(0,v) \pm g_2(0,v)\sqrt{f_3(0,v)} = 0, \;\; g_1(1,v) \pm g_2(1,v)\sqrt{f_3(1,v)} = 0, \;\;0 \leq v \leq 1\;,$     (8.68)

which can be solved by the same auxiliary variable method as before.

When $ f_3 = 0$ (or equivalently $ H^2 - K = 0$ if $ S \neq 0$ ), (8.60) become singular. This condition is equivalent to the point where the two principal curvatures are identical, i.e. an umbilical point. If the umbilical point coincides with a local maximum or minimum of the curvature, we cannot use (8.66) to locate such a point. In such case we need to locate the umbilical point first by finding the roots of the equation

$\displaystyle H^2(u,v) - K(u,v) = \frac{f_3(u,v)}{4S^6(u,v)} = 0\;,$     (8.69)

which reduces to solving $ f_3(u,v) = 0$ , since $ S \neq 0$ . Because $ W(u,v)\equiv\frac{f_3(u,v)}{4S^6(u,v)}$ is a non-negative function, $ W(u,v)$ has a global minimum at the umbilic [257]. The condition for global minimum at the umbilic implies that $ \nabla W = {\bf0}$ or equivalently (given that $ f_3(u,v) = 0$ )
$\displaystyle W_u = \frac{\frac{\partial f_3}{\partial u}}{4S^6} = 0, \;\;\; W_v = \frac{\frac{\partial f_3}{\partial v}}{4S^6}= 0\;.$     (8.70)

Therefore, assuming $ S \neq 0$ , the umbilics are the solutions of the following three simultaneous equations (see also Sect. 9.3):
$\displaystyle \frac{\partial f_3(u,v)}{\partial u} = 0, \;\;\; \frac{\partial f_3(u,v)}{\partial v} = 0, \;\;\; f_3(u,v) = 0, \quad \quad 0 \leq u,v \leq 1\;.$     (8.71)

These equations can be reduced to [257]:
$\displaystyle BB_u - 2A_uS^2 - 4A({\bf S}\cdot{\bf S}_u) = 0, \;\;\; BB_v - 2A_vS^2 - 4A({\bf S}\cdot{\bf S}_v) = 0\;,$      
$\displaystyle B^2 - 4AS^2 = 0, \;\;\; 0 \leq u,v \leq 1\;.$     (8.72)

Since $ f_3(u,v) = 0$ at the umbilics, (8.66) reduce to $ f_1(u,v) = 0$ , $ g_1(u,v) = 0$ . If we substitute the first equation of (8.72) into (8.61) and use the fact $ f_3 = B^2 - 4AS^2 = 0$ , we obtain $ f_1(u,v) = 0$ . Similarly by substituting the second equation of (8.72) into (8.64), we obtain $ g_1(u,v) = 0$ . Consequently the solutions of (8.66) include not only the locations of extrema of principal curvatures but also the locations of the umbilical points. Then we use Theorem 9.5.1 at the umbilical points to check if the umbilical point is a local extremum of principal curvatures.

A cusp is an isolated singular point on the surface where the surface tangent plane is undefined, i.e. $ {\bf r}_u \times {\bf r}_v = {\bf0}$ . Cusps of an offset surface correspond to points on the progenitor where both of the principal curvatures are equal to $ -\frac{1}{d}$ where $ d$ is the offset distance [94] (see also Sect. 11.3.2). In this manner, cusps on an offset surface are associated with umbilics of the progenitor. Hence we can locate the cusps on an offset surface using (8.71).

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December 2009