5.5.1 Point/implicit algebraic surface intersection

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The mathematical description for a point and an implicit surface $f(\mathbf r)=0$ intersection is given by

$\displaystyle \mathbf{r}_0 \cap f(\mathbf{r})=0\;.$

(5.31)

Similar to the point implicit curve intersection case, the condition for intersection is given by

$\displaystyle \vert f(\mathbf{r}_0)\vert < \varepsilon, \quad \frac{\vert f(\mathbf{r}_0)\vert}{\vert\nabla f(\mathbf{r}_0)\vert} < \delta\;,$

(5.32)

where $\varepsilon, \delta$ are small positive constants, provided $\vert\nabla f(\mathbf{r}_0)\vert\neq 0$ .

December 2009