8.1.3.4 Surface curvatures and curvature maps

Surface interrogation may be performed using different curvature measures of the surface [22,280,18,92]. The Gaussian, mean, absolute [92], and root mean square (rms) [246] curvatures are the product (3.61), the average (3.62), the sum of the absolute values

and the square root of the sum of the squares of the principal curvatures

respectively. These surface curvature functions are all scalar valued functions. The variation of any of these quantities can be displayed using a color-coded curvature map [74,22]. Contour lines of constant curvature can also be used to display and visualize the variation of these curvature functions. Munchmeyer [280] calculates the curvature on a lattice and linearly interpolates the contour points. Maekawa and Patrikalakis [255] present a method which allows a robust and accurate computation of all stationary points and all contour lines for functions describing Gaussian, mean and principal curvatures of B-spline surfaces. This method allows us to divide the surface into regions of specific range of curvature. A summary with further applications can be found in [255].

A surface inflection exists on a surface at a point if the surface crosses the tangent plane at [172]. The point is then called an inflection point. If the Gaussian curvature is positive at every point of a region of a surface then there are no inflections in that region. If the Gaussian curvature changes sign in a region of a surface then there is an inflection in that region. In areas where the surface has Gaussian curvature very close to or equal to zero the Gaussian curvature alone cannot provide adequate information about the shape of the surface. In such a case the surface has an inflection point in the region only if the mean curvature changes sign. The Gaussian and mean curvatures together provide sufficient information in order to identify surface inflections on a surface [279,280].

The curvature maps of the principal curvatures can also help to select a spherical cutter of suitable radius in order to avoid gouging during machining of the surface [22,255]. We will discuss further on how to construct contour lines of constant curvature in Sect. 8.5.