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Journal for Geometry and Graphics 18 (2014), No. 2, 159--172
Copyright Heldermann Verlag 2014

Skinning of Circles and Spheres by Geometric Optimization in Minkowski Space

Bernhard Blaschitz
Waagner-Biro Stahlbau AG, Leonard-Bernstein-Str. 10, 1220 Wien, Austria

Assuming a discrete set of circles pi 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 pi and a cubic B-Spline b by observing the footpoint problem, which brings a better fit, but results in a non-linear problem. The reality of the envelope results in a quadratic, but non-convex 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 1-parameter case that leads to a new method for interpolation points in the Minkowski space R3,1 by curves, which translates to interpolation of spheres by canal surfaces. Approximating 2-parameter sets of points by surfaces in the Minkowski space R3,1 gives rise to general envelope surfaces of 2-parameter families of spheres, that have not been studied before in this generality. For this, a calculus was reinvented and applied, that classifies 2-planes in R3,1 according to their steepness.

Keywords: Minkowski space, numerical optimization, curve fitting, surface fitting, Laguerre geometry, cyclography.

MSC: 51B20; 68U07, 51N20, 51N30, 65K10, 74P20

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