Page 212 - Geometric Modeling and Algebraic Geometry

12

Approximate Implicitization of Space Curves
and of Surfaces of Revolution

Mohamed Shalaby and Bert J¨uttler

Institute of Applied Geometry,
Johannes Kepler University, Linz, Austria
bert.juettler@jku.at
Summary. We present techniques for creating an approximate implicit representation of
space curves and of surfaces of revolution. In both cases, the proposed techniques reduce the
problem to that of implicitization of planar curves. For space curves, which are described as
the intersection of two implicitly defined surfaces, we show how to generate an approximately
orthogonalized implicit representation. In the case of surfaces of revolution, we address the
problem of avoiding unwanted branches and singular points in the region of interest.

12.1 Introduction

Traditionally, most CAD (Computer Aided Design) systems rely on piecewise ratio-
nal parametric representations, such as NURBS (Non–Uniform Rational B–Spline)
curves and surfaces. The parametric representation offers a number of advantages,
such as simple sampling techniques, which can be used for quickly generating an ap-
proximating triangulation for visualization. On the other hand, the use of implicitly
defined curves/surfaces also offers a number of advantages, e.g., for solving inter-
section problems, or for visualization via ray–tracing.
In order to exploit the potential benefits of using the implicit representation of
curves and surfaces, methods for conversion from parametric to implicit form (im-
plicitization) are needed. As an alternative to exact methods, such as resultants,
Gr¨ obner bases, moving curves and surfaces, etc. [2, 4, 5, 8, 14], a number of ap-
proximate techniques have emerged [3, 7, 10, 11]. As demonstrated in the frame of
the European GAIA II project [6, 15, 17], these techniques are well suited to deal
with general free–form curve and surface data arising in an industrial environment.
On the other hand, CAD objects typically involve many special curves and sur-
faces, such as natural quadrics, sweep surfaces, surfaces of revolution, etc. While
implicit representations of simple surfaces are readily available, this paper studies
approximate approximation of two special objects, namely space curves and surfaces
of revolution. Space curves arise frequently in geometric modeling. An implicit rep-
resentation of a space curve is given by the intersection of two implicitly defined