Curve and Surface Interrogation

Abstract

Interrogation is the process of extraction of information from a geometric model. In this chapter we focus on free-form curve and surface interrogation. Free form surface, also called sculptured surface, are widely used in scientific and engineering applications. For example, the hydrodynamic shape of propeller blades has an important role in marine applications, and the aerodynamic shape of turbine blades determines the performance of aircraft engines. Free-form surfaces arise also in the bodies of the ships, automobiles and aircraft, which have both functionality and attractive shape requirements. Many electronic devices as well as consumer products are designed with aesthetic shapes, which involve free-form surfaces. During the last two decades, many curve and surface interrogation methods have been proposed (see references in [275, 61]), and we will introduce some of them in this chapter. Propeller and turbine blades are manufactured by numerically controlled (NC) milling machines. When a ball-end mill cutter is used, the cutter radius must be smaller than the smallest concave radius of curvature of the surface to be machined to avoid local overcut (gouging) (see Sect. 11.1.2). Gouging is the one of the most critical problems in NC machining of free-form surfaces [184]. Therefore, we must determine the distribution of the principal curvatures of the surface, which are upper and lower bounds on the curvature at a given point, to select the cutter size [116, 94]. Visualization techniques of various curvature measures have been developed by Dill [74], Beck et al. [22], Munchmeyer [280, 279], Higashi and Kaneko [162], Pottmann and Optiz [328], Maekawa and Patrikalakis [255], Elber and Cohen [87] and Tuohy [423]. Higashi et al. [163] introduced the loci of points corresponding to extrema of curvature values of the design surface called surface edges, which show how the surface is waving and where the peaks of the wave exist. Kase et al. [189] presented local and global evaluation methods for shape errors of free-form surfaces which have been applied to the evaluation of sheet metal surfaces.