5.8.2 Rational polynomial parametric/rational polynomial parametric surface intersection (Case F1)

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5.8.2 Rational polynomial parametric/rational polynomial parametric surface intersection (Case F1)

Rational polynomial parametric surface to rational polynomial parametric surface intersection is defined as:

$\displaystyle {\bf r} = \mathbf{r}_1(\sigma,t) = \left(\frac{X_1(\sigma,t)}{W_1... ...)},\frac{Z_1(\sigma,t)}{W_1(\sigma,t)} \right)^T,\;\; 0 \leq \sigma, t\leq 1\;,$			(5.99)
$\displaystyle \cap\;\;{\bf r} = \mathbf{r}_2(u,v) = \left(\frac{X_2(u,v)}{W_2(u... ...2(u,v)}{W_2(u,v)},\frac{Z_2(u,v)}{W_2(u,v)} \right)^T,\;\; 0 \leq u,v \leq 1\;.$

Formulation can be provided by setting $\mathbf{r}_1(\sigma,t)=\mathbf{r}_2(u,v)$ which leads to three nonlinear polynomial equations for four unknowns $\sigma, t, u, v$ . It is an underconstrained system with 3 equations and 4 unknowns. This system can be solved by the IPP algorithm of Chap. 4. However, as the solutions are typically not isolated points but curves, such approach is very slow when small tolerances are used. One could also implicitize $\mathbf{r}_1(\sigma,t)$ to the form and substitute $x=\frac{X_2(u,v)}{W_2(u,v)}$ , $y=\frac{Y_2(u,v)}{W_2(u,v)}$ and $z=\frac{Z_2(u,v)}{W_2(u,v)}$ into to reduce the problem to Case F3 for low degree surfaces [212]. Heo et al. [160] studied the intersection of two ruled surfaces which is simpler than the general parametric surface to surface intersection problem.

There are three major techniques for solving RPP/RPP surface intersections. Detailed reviews can be found in [295,300].

Subsections

Next: 5.8.2.1 Lattice methods Up: 5.8 Surface/surface intersections Previous: 5.8.1.5 Computing starting points Contents Index

December 2009