Hybrid Symbolic and Numeric Operators

				   as

		 Tools for Analysis of Freeform Surfaces




		     Gershon Elber and Elaine Cohen



			       UUCS-92-023


		     Department of Cumputer Science
			   University of Utah
		     Salt Lake City, UT 84112, USA


			        Abstract

Freeform surfaces are commonly used in Computer Aided Geometric
Design, so accurate analysis of surface properties is becoming
increasingly important. In this paper we define surface slope and
surface speed, develop visualization tools, and demonstrate that they
can be useful in the design process.  Generally, surface properties
such as curvature and twist are evaluated at a finite set of
predetermined samples on the surface.  This paper takes a different
approach.  A small set of tools is used to symbolically compute
surfaces representing curvature, twist and other properties. These
surfaces are then analyzed using numeric techniques.

	The combination of symbolic computation to provide an exact
property representation (up to machine accuracy) and numerical methods
to extract data is demonstrated to be powerful and robust. This
approach supports a uniform treatment once the surfaces are computed
and also provides global information, so questions such as `is a
surface developable?' or `what are the hyperbolic regions of a
surface?' can be answered robustly.