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8.5.2 Finding starting points

Contour lines in the parameter space of a bivariate function can be separated into three categories: If the surface is subdivided along the iso-parametric lines which contain the local maxima and minima of curvature inside the domain and the contouring levels of curvature are chosen such that the contour curves avoid saddle points, as shown in Figs. A.3, A.4, A.5, A.6, each sub-patch will contain simple contour branches without loops or singularities. Therefore we can find all the starting points of the various levels of contour curves along the parameter domain boundary of each sub-patch by finding the roots of the following equations.

Starting with the Gaussian curvature

$\displaystyle K(u,0) = \frac{A(u,0)}{S^4(u,0)} = C_{K},\; K(u,1) = \frac{A(u,1)}{S^4(u,1)} = C_{K},\;0\leq u \leq 1\;, \;\;\;\;\;\;\;\;\;$     (8.111)
$\displaystyle K(0,v) = \frac{A(0,v)}{S^4(0,v)} = C_{K},\; K(1,v) = \frac{A(1,v)}{S^4(1,v)} = C_{K},\; 0 \leq v \leq 1\;,\;\;\;\;\;\;\;\;\;$     (8.112)

where $ C_K$ is the constant Gaussian curvature value. These equations can be rewritten as follows:
$\displaystyle C_KS^4(u,0) - A(u,0) = 0, \;C_KS^4(u,1) - A(u,1) = 0, \;0 \leq u \leq 1\;, \;\;\;\;\;$     (8.113)
$\displaystyle C_KS^4(0,v) - A(0,v) = 0, \;\; C_KS^4(1,v) - A(1,v) = 0, \;\; 0 \leq v \leq 1\;.\;\;\;\;\;$     (8.114)

Equations (8.113), (8.114) are univariate polynomials of degree $ 8m$ -$ 4$ and $ 8n$ -$ 4$ , respectively.

Similarly for the mean curvature

$\displaystyle H(u,0) = \frac{B(u,0)}{2S^3(u,0)} = C_{H}, \; H(u,1) = \frac{B(u,1)}{2S^3(u,1)} = C_{H}, \; 0 \leq u \leq 1\;,\;\;\;\;\;\;\;\;$     (8.115)
$\displaystyle H(0,v) = \frac{B(0,v)}{2S^3(0,v)} = C_{H}, \; H(1,v) = \frac{B(1,v)}{2S^3(1,v)} = C_{H}, \; 0 \leq v \leq 1\;, \;\;\;\;\;\;\;\;$     (8.116)

where $ C_H$ is the constant mean curvature value. These equations can be rewritten as follows
$\displaystyle B(u,0) - 2C_H\sqrt{S^2(u,0)}S^2(u,0) = 0, \;\;\; B(u,1) - 2C_H\sqrt{S^2(u,1)}S^2(u,1) = 0\;,$     (8.117)
$\displaystyle B(0,v) - 2C_H\sqrt{S^2(0,v)}S^2(0,v) = 0, \;\;\; B(1,v) - 2C_H\sqrt{S^2(1,v)}S^2(1,v)= 0\;,$     (8.118)

where $ 0\leq u,v \leq 1$ . Equations (8.117), (8.118) are the univariate irrational functions involving polynomials and square roots of polynomials which come from the normalization of the normal vector of the surface (see (3.3)). $ B(u,v)$ is a polynomial of degree $ (5m-3, 5n-3)$ and $ S^2(u,v)$ is a polynomial of degree $ (4m-2, 4n-2)$ .

Finally for the principal curvatures

$\displaystyle \kappa(u,0) = \frac{B(u,0) \pm \sqrt{f_3(u,0)}}{2S^3(u,0)} = C_{\kappa}\;,$     (8.119)
$\displaystyle \kappa(u,1) = \frac{B(u,1) \pm \sqrt{f_3(u,1)}}{2S^3(u,1)} = C_{\kappa}\;,$     (8.120)
$\displaystyle \kappa(0,v) = \frac{B(0,v) \pm \sqrt{f_3(0,v)}}{2S^3(0,v)} = C_{\kappa}\;,$     (8.121)
$\displaystyle \kappa(1,v) = \frac{B(1,v) \pm \sqrt{f_3(1,v)}}{2S^3(1,v)} = C_{\kappa}\;,$     (8.122)

where $ 0\leq u,v \leq 1$ , the $ \pm $ signs correspond to the maximum and minimum principal curvatures, and $ C_{\kappa}$ is the constant value of principal curvature and $ f_3(u,v)$ is a polynomial function defined in (8.63). Equations (8.119) through (8.122) can be rewritten as follows:
$\displaystyle B(u,0) \pm \sqrt{f_3(u,0)} -2C_{\kappa}S^2(u,0)\sqrt{S^2(u,0)} = 0,\;\;\; 0 \leq u \leq 1\;,$     (8.123)
$\displaystyle B(u,1) \pm \sqrt{f_3(u,1)} -2C_{\kappa}S^2(u,1)\sqrt{S^2(u,1)} = 0, \;\;\; 0 \leq u \leq 1\;,$     (8.124)
$\displaystyle B(0,v) \pm \sqrt{f_3(0,v)} -2C_{\kappa}S^2(0,v)\sqrt{S^2(0,v)} = 0, \;\;\; 0 \leq v \leq 1\;,$     (8.125)
$\displaystyle B(1,v) \pm \sqrt{f_3(1,v)} -2C_{\kappa}S^2(1,v)\sqrt{S^2(1,v)} = 0, \;\;\; 0 \leq v \leq 1\;.$     (8.126)

Equations (8.123) through (8.126) are the univariate irrational functions involving polynomials and two square roots of polynomials which come from the analytical expression of the principal curvature and normalization of the normal vector of the surface. The starting points for contour curves of curvature occur in pairs, since non-loop contour lines must start from a domain boundary and must end at a domain boundary point.

Cuspidal edges are loci of points on the surface which have tangent discontinuity and appear as sharp edges on the surface. Cuspidal edges of an offset surface correspond to points on the progenitor where one of the principal curvatures is equal to $ -\frac{1}{d}$ ($ d$ is the offset distance) [94]. Therefore we can use (8.123) through (8.126) to compute the starting points for tracing cuspidal edges on the offset, if the surface is subdivided along the iso-parametric lines which contain the local maxima and minima of principal curvatures, such that each sub-patch will not contain loops of cuspidal edges in its interior.

next up previous contents index
Next: 8.5.3 Mathematical formulation of Up: 8.5 Contouring constant curvature Previous: 8.5.1 Contouring levels   Contents   Index
December 2009