Page 159 - Geometric Modeling and Algebraic Geometry

9

Intersecting Biquadratic B´ ezier Surface Patches

St´ ephane Chau , Margot Oberneder , Andr´ e Galligo ,and BertJ¨uttler 2
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Laboratoire J.A. Dieudonn´ e, Universit´ e de Nice - Sophia-Antipolis, France
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{chaus,galligo}@math.unice.fr
Institute of Applied Geometry, Johannes Kepler University, Austria
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{margot.oberneder,bert.juettler}@jku.at
Summary. We present three symbolic–numeric techniques for computing the intersection
and self–intersection curve(s) of two B´ ezier surface patches of bidegree (2,2). In particular,
we discuss algorithms, implementation, illustrative examples and provide a comparison of the
methods.

9.1 Introduction

The intersection of two surfaces is one of the fundamental operations in Computer
Aided Design (CAD) and solid modeling. Closely related to it, the elimination of
self–intersections (which may arise. e.g., from offsetting) is needed to maintain the
correctness of a CAD model. Tensor–product B´ ezier surface patches, which are para-
metric surfaces defined by vector–valued polynomials x :[0, 1] → 3 of certain
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bidegree (m, n), are extensively used to model surfaces in CAD and solid model-
ing. However, even for relatively small bidegrees m, n ≤ 3, the intersection and
self–intersection loci of such patches can be fairly complicated. Consequently, stan-
dard algorithms for surface–surface intersections [24, 28] generally do not take the
properties of special classes of such tensor–product surfaces into account.
In the case of two general surfaces, a brute–force approach to compute the inter-
section curve(s) consists in (step 1) approximating the surface by triangular meshes
and (step 2) intersecting the planar facets of these meshes. Clearly, in order to achieve
high accuracy, a very fine approximation with a mesh may be needed. Alternatively,
one may consider to choose another, more complicated representation, where the ba-
sic elements are capable of capturing more of the geometric features. For instance,
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one may choose quadratic triangular patches or biquadratic tensor–product patches .
Clearly, this approach would need robust intersection algorithms for the more com-
plicated basic elements.

In the same spirit, Reference [32] proposes to use triangular patches for efficient visualiza-
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tion.