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Journal for Geometry and Graphics, Vol. 2, No. 2, pp. 109 - 121 (1998)
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#
Edge-Orthogonal Patches through a Given Rational Bézier Curve

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Johann Lang, Hans-Peter Schröcker

Institute for Geometry, Graz University of Technology

Kopernikusgasse 24, A-8010 Graz, Austria

email: lang@geometrie.tu-graz.ac.at

**Abstract:** Applications in computational fluid dynamics (CFD) have led to the problem of finding a rational Bézier patch with a given edge parameter line $k$ the way that the parameter lines of the other type intersect $k$ orthogonally. This is what we call an orthogonal continuation of $k$. The variety of solutions to the problem is being investigated and a very geometric way for the construction of the solutions is being offered. Using some fundamental features of polynomials we can establish a link between the properties of the weight polynomial and the elevation of degree which is necessary to find non-trivial orthogonal continuations. For some cases which turn out to be unsolvable, and for cases where the solution existing has a very high degree, we can describe a Monte Carlo method providing surprisingly good approximations. This method is even capable of coping with tasks where the right angle is replaced by some arbitrary angle function.

**Keywords:** Bézier surfaces, orthogonal continuation, Monte Carlo method

**Classification (MSC2000):** 53A05; 68U05

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