
Journal for Geometry and Graphics 18 (2014), No. 2, 159172 Copyright Heldermann Verlag 2014 Skinning of Circles and Spheres by Geometric Optimization in Minkowski Space Bernhard Blaschitz WaagnerBiro Stahlbau AG, LeonardBernsteinStr. 10, 1220 Wien, Austria bernhard.blaschitz@waagnerbiro.at Assuming a discrete set of circles p_{i} in the plane, a real envelope is looked for. The new approach of this work is reformulating the original task as a constrained optimization in the point set model. The quadratic objective function minimizes the Euclidean distance between the cyclographic images of circles p_{i} and a cubic BSpline b by observing the footpoint problem, which brings a better fit, but results in a nonlinear problem. The reality of the envelope results in a quadratic, but nonconvex constraint, which can be linearized. This linearization is discussed in detail, as its formulation is central to this work. The ideas discussed for circles are also generalized for spheres; in the 1parameter case that leads to a new method for interpolation points in the Minkowski space R^{3,1} by curves, which translates to interpolation of spheres by canal surfaces. Approximating 2parameter sets of points by surfaces in the Minkowski space R^{3,1} gives rise to general envelope surfaces of 2parameter families of spheres, that have not been studied before in this generality. For this, a calculus was reinvented and applied, that classifies 2planes in R^{3,1} according to their steepness. Keywords: Minkowski space, numerical optimization, curve fitting, surface fitting, Laguerre geometry, cyclography. MSC: 51B20; 68U07, 51N20, 51N30, 65K10, 74P20 [ Fulltextpdf (5505 KB)] for subscribers only. 