Note on curve and surface energies

Energies of curves and surfaces together with their discrete variants play a prominent role as fairness functionals in geometric modeling and computer aided geometric design. This paper deals with a particular discrete surface energy which is expressible in terms of curve energies, and which occurs naturally in the problem of smoothing digital elevation data with tolerance zone constraints. We also discuss geometrically meaningful surface energies in general from the viewpoint of invariant theory, and the role of the Gauss-Bonnet theorem. 1. A FAIRING PROBLEM WITH HARD CONSTRAINTS For a the polyline p with vertices pi we consider the discrete linearized bending energy (1) E2(p) = ∑‖∆pi‖, where the forward difference operator ∆ is defined by ∆pi = pi+1− pi, which implies ∆pi = pi+2− 2pi+1+ pi. This energy functional occurs e.g. in the context of energy-minimizing curves and curve networks (cf. Hofer and Pottmann (2004) and Wallner et al. (2005)). Hofer et al. (2005) apply energy-minimizing curve networks to the problem of smoothing digital elevation data, as described below. Suppose that a terrain is modeled by a rectangular grid of points pi j = (xi j,yi j,zi j), with i = 0, . . . ,N and = 0, . . . ,M. The x and y coordinates have the form (2) xi j = i ·δx, yi j = j ·δy (δx,δy > 0). So in fact the shape of the terrain is given only by the height field zi j. Terrain smoothing according to Hofer et al. (2005) starts with digital terrain data, i.e., a height field. Each vertex pi j is the center of its tolerance cylinder Zi j whose axis is parallel to the z-axis, and whose diameter and height typically is of the same magnitude as distances between vertices (see Fig. 1a). The problem dealt with in that paper is to compute a new height field zi j which is smoother than the original and describes a surface which passes through each tolerance cylinder. The latter condition is certainly fulfilled if pi j = (xi j,yi j,z ′ i j) ∈ Zi j, as visualized by Fig. 1b. But there is a weaker condition which still implies the required property of the new height field: It is sufficient each cylinder is met either by a ‘horizontal’ or a ‘vertical’ grid lines. In other words, it is sufficient that for all i, j, the cylinder Zi j must intersect at least one of the four line segments pi jp ′ i−1, j, p ′ i jp ′ i+1, j, p ′ i jp ′ i, j−1, p ′ i jp ′ i, j+1. It is easy to check if this condition is met. It is illustrated in Fig. 1c. Starting from a given terrain (Fig. 2a), a fairness functional is minimized such that the constraints described above are respected. An example of the result of such a process is shown by Fig. 2. 1