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5.7.1 Rational polynomial parametric curve/implicit algebraic surface intersection (Case E3)

The intersection problem is defined as:
$\displaystyle {\bf r}= {\bf r}(t)=\left(\frac{X(t)}{W(t)}, \frac{Y(t)}{W(t)}, \frac{Z(t)}{W(t)}\right)^T \; \cap \; f({\bf r})=0, \quad 0\leq t \leq 1\;.$     (5.68)

Let us consider an implicit algebraic surface of total degree $ m$
$\displaystyle f(x,y,z) = \sum_{i=0}^{m} \sum_{j=0}^{m-i} \sum_{k=0}^{m-i-j} c_{ijk}x^i y^j z^k = 0\;.$     (5.69)

We substitute $ x = \frac{X(t)}{W(t)} $ , $ y = \frac{Y(t)}{W(t)}$ and $ z = \frac{Z(t)}{W(t)} $ of degree $ n$ into the implicit equation and multiply by $ W^m(t) $ leading to
$\displaystyle F(t) = \sum_{i=0}^{m} \sum_{j=0}^{m-i} \sum_{k=0}^{m-i-j} c_{ijk} X^i(t) Y^j(t) Z^k(t) W^{m-i-j-k} (t)=0\;,$     (5.70)

of degree $ \leq mn$ in $ t$ . We then find its real roots in $ [0, 1]$ , as described in Sect. 5.6.1.

Alternatively, the problem can be formulated as a nonlinear polynomial system of four equations in four unknowns ($ x$ , $ y$ , $ z$ , $ t$ ) and solved using the IPP algorithm.

next up previous contents index
Next: 5.7.2 Rational polynomial parametric Up: 5.7 Curve/surface intersection Previous: 5.7 Curve/surface intersection   Contents   Index
December 2009