Scale-Invariant Functional for Smooth Curves and Surfaces

Abstract

Various functionals for optimizing the fairness of curves and surfaces are compared. Minimizing these functionals leads to the well-known Minimum Energy Curve (MEC) and Minimum Energy Surface (MES), to the more recently discussed Minimum Variation Curve and Surface (MVC and MVS), and to their scale-invariant (SI-) versions. The use of a functional that minimizes curvature variation rather than bending energy leads to shapes of superior fairness and, when compatible with any external interpolation constraints, forms important geometric modeling primitives: circles, helices, and cyclides (spheres, cylinders, cones, and tori). The addition of the scale-invariance property leads to stability and to the possibility of studying curves and surfaces that are determined only by their topological shape, free of any external geometrical constraints. The behavior of curves and surfaces optimized with the different functionals is demonstrated and discussed on simple representative examples. Optimal shapes for curves of various turning numbers and for some low-genus surfaces are presented.