Re: Intersecting curved patched

On Fri, 02 Sep 2005 00:53:09 +0200, Wolfgang Draxinger
<wdraxinger@xxxxxxxxxxxxxxxx> wrote:
>The intersection of curved patches are curved lines, right? Can
>the parametrization or an approximitation of these curves be
>deterimined either analytically or numerically, if is so, how is
>it done?

Yes, the intersection of two cubic Bezier patches (or other parametric
polynomials, such as B-spline patches) is a polynomial curve. However,
by Bezout's theorem, the degree is quite high: each patch has implicit
degree 18 (=2*3*3), so the curve has degree 324 (=18*18). Ouch.

In practical terms, this forces the use of approximations. One option
is to trace a curve in the parametric space of a patch; another is to
trace a curve in space. Tracing a curve requires finding a point of
intersection, then using numerical methods to step to nearby points.
Often there will be multiple disjoint pieces, sometimes well separated
and sometimes not. All these features make the problem a challenge.

There is a body of professional literature, since the CAD industry
needs to do this in commercial design software. A good place to start
might be the presentation Sederberg gave for his colleagues in 2002:

"Surface-Surface Intersections: Robust Topological Methods"
<http://www.darpa.mil/dso/thrust/math/cargo/reviews/may2002.htm>

Another recent discussion is

"A Practical Algorithm for Surface/Surface Intersection"
<http://www.cs.berkeley.edu/~hling/research/paper/intersection.htm>

Beyond that, try a web search for "surface-surface intersection".

.

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