On the Curvature of Curves and Surfaces Defined by Normalforms

Erich Hartmann

(Comp. Aided Geom. Des. 16, 355-376)

Abstract:
The normalform h=0 of a curve (surface) is a generalization of the Hesse normalform of a line in (plane in ). It was introduced and applied to curve and surface design in recent papers. For determining the curvature of a curve (surface) defined via normalforms it is necessary to have formulas for the second derivatives of the normalform function h depending on the unit normal and the normal curvatures of three tangential directions of the surface . These are derived and applied to visualization of the curvature of bisectors and blending curves, isophotes, curvature lines, feature lines and intersection curves of surfaces. The idea of the normalform is an appropriate tool for proving theoretical statements, too. As an example a simple proof of the Linkage Curve Theorem is given.

Keywords: normalform, Hessian matrix, curvature, normal curvature, bisector, -blending, -continuity, umbilic points, isophote, curvature line, feature line, ridge, ravine, intersection curve, foot point,

 
Figure 1: Isophotes of a -blending of two tensor product Bézier surfaces