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.
Developable surfaces [13,222,120,32,326,252,329] are surfaces which can be unfolded or developed onto a plane without stretching or tearing. They are of considerable importance to plate-metal-based industries as shipbuilding. For a developable surface the Gaussian curvature is zero everywhere [76]. Thus the manufacturer would profit from prior knowledge of the distribution of the Gaussian curvature of the metal plate. A line of curvature indicates a directional flow for the maximum or the minimum curvature across the surface [22,279,173,257,251,98] which can be used for determining feeding directions to the rolling machine for the metal plate.
Fairing [365,339,144,318,148,45,317] is the process of eliminating shape irregularities in order to produce a smoother shape. Reflection lines [202,190,61] are a standard surface interrogation method to assess the fairness of design surfaces in the automobile industry. Isophotes are used for detection of surface irregularities [319,144] and for continuity evaluation at the boundaries of adjacent patches [146]. Also focal surfaces [146,145] are used to detect undesired curvature properties of a design surface. The set of curvature extrema of a fair surface should coincide with the designer's intention. Therefore, computation of all extrema of curvatures is desirable. The Gaussian, mean and principal curvatures are used for the detection of surface irregularities [280,279,255]. On the other hand Andersson [7] developed a method to specify curvature in a surface design method and Higashi et al. [164] proposed a method to generate a smooth surface by controlling the curvature distribution. Theisel and Farin [419] studied the curvature of characteristic curves on a surface, such as contour lines, lines of curvature, asymptotic lines, isophotes and reflection lines.
As we will see in this chapter, the governing equations for shape interrogation often result in polynomial equations with unknowns when the input curves and surfaces are in integral/rational Bézier form. If the parametric curves and surfaces are in integral/rational B-spline form, some shape interrogation methods involve a first step where the integral/rational B-spline curve or surface patch is subdivided into a number of integral/rational Bézier curves or rational Bézier patches. In the computer implementation we evaluate the coefficients of the governing nonlinear polynomial equations in multivariate Bernstein form starting from the given input Bézier curve or surface using the arithmetic operations in Bernstein form [106,315] (see Sect. 1.3.2). The system of nonlinear polynomial equations can be solved robustly and accurately by the IPP algorithm presented in Chap. 4.
In this section we will classify the interrogation methods by the order of derivatives of the curve or surface position vector which are involved. An interrogation method is characterized as nth-order, if derivatives of the curve or surface position vector of order n are involved.