TFRWORTY EINEMANN Computer-Aided Design, Vol. 27. NO. 7, pp. 523543.1995 EIwvierScienceLtd Printed in Great Britain 0010-4495195 $10.00+0.00 Simplified curve and surface interrogation via mathematical packages and graphics libraries and hardware Henry P Moreton During the process of designing smoothly curved forms, it is important to have methods available for the assessment of shape quality. The advent of computer graphics has made it possible to perform curve and surface interrogation without the construction of a physical model. Methods have relied on renderings, starting with wireframe representations, and in- creasing in sophistication to today’s highly realistic render- ings. Also, interrogation techniques have been developed to exaggerate shape, making it easier to detect a curve’s or surface’s subtler aspects or flaws. The use of interrogation techniques is not limited to curve and surface design. They can be used to enhance the display of curve and surface data from many sources. The paper presents traditional and effective new methods for curve and surface interrogation. The implementation of these methods is simplified by exploiting mathematics pack- ages and graphics library and hardware support. Keywords: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA surface, analysis, interrogation The design of smoothly curved shapes on computers has become common practice. Because the design process proceeds without the creation of a physical model, methods have had to be developed that allow a designer to evaluate shape quality without actually being able to touch or feel the objects themselves. We refer to these methods collectively as shape interroga- tion techniques. Recent advances in ‘3D printing’ may facilitate the evaluation process by quickly creating physical models, but, because of the highly interactive nature of the design process, visually based methods for shape interrogation will remain important. Interrogation techniques attempt to illuminate curve and surface characteristics that are not easily dis- cernible using conventional rendering. These character- istics are usually intrinsic geometric properties such as Silicon zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Graphics Inc., MountainView, CA 94039-7311, USA zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Paper received: 10 July 1994 curvature and torsion. There are applications in which representation-specific properties such as parameteri- zation are important and must also be conveyed to the designer. Many of today’s computer graphics workstations come with specialized hardware to accelerate common rendering activities such as Gouraud shading, specular lighting, hidden surface removal, antialiasing, and tex- ture and environment mapping. In addition, these workstations also have graphics library support for the drawing of a variety of primitives, lines, polygons, curves and surfaces, spheres, cylinders, etc. By taking advan- tage of these facilities, it is possible to implement techniques for curve and surface interrogation more easily. Curves and surfaces represented on a computer can be described using a variety of methods which can be broadly classified as implicit, explicit, and parametric. The most commonly used representations are based on vector valued parametric polynomials, such as NURBS. Although the methods discussed in this paper are ap- plicable to most representations, we will restrict our discussion to parametric curves and surfaces. The remainder of this paper is organized as follows. In the second section, we review the properties of curves and surfaces and their representations. In the third section, we survey interrogation techniques used in curve and surface design and evaluation. In the fourth section, we discuss the use of mathematics pack- ages, and the graphics hardware and software used in the creation of the figures in this paper. PROPERTIES OF CURVES AND SURFACES In this section*,,we review the geometric properties of curves and surfaces as well as the terminology used for these properties. The material covered in this section can also be found in texts such as References 2 and 3. *Portions of this section are taken from Reference 1. Computer-Aided Design Volume 27 Number 7 July 1995 523