Discrete Nonlinear Fairing of Curves and Surfaces

A popular technique to simplify this approach is to give up the parameter independence by approximating the geometric invariants with higher order derivatives.For some important fairness functionals this results in algorithms that enable the construction of a solution by solving a linear equation system, but such curves and surfaces are in most cases not as fair as those depending on geometric invariants only.

An interesting approach to simplify the construction process without giving up the geometric intrinsics is to use variational calculus to derive differential equations characterizing the solution of a minimization problem. The usage of such differential equations based on geometric invariants can be seen as a reasonable approach to the fairing problem in its own right and it can be applied to curves as well as for surfaces. For planar curves one of the simplest differential equations is $\kappa''=0$. Assuming arc length parameterization this equation is only satisfied by lines, circles and clothoids. A curvature continuous curve that consists of parts of such elements is called a clothoid spline. Most algorithms for the construction of such curves are based on techniques that construct the according line, circle and clothoid segments of the spline. This is possible for planar curves, but the idea does not extend to surfaces.


In this talk we will present a fast algorithm to construct an interpolating discrete clothoid spline (DCS) purely based on its characteristic differential equation. Our algorithm uses discrete data, because this has been proven to be especially well suited for the efficient construction of nonlinear splines. The efficiency of our construction process is largely based on an algorithm called indirect approach. This algorithm decouples the curvature information from the actual geometry by exploiting the fact that the curvature distribution itself is a discrete linear spline. Beside the speed of the planar algorithm, it has another very important property: it directly extends to the construction of surfaces!