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Applications of sphere geometries in computational design
Helmut Pottmann
KAUST, Saudi Arabia
The classical sphere geometries of Möbius, Laguerre and Lie provide a rich source of knowledge which can be highly useful in the solution of problems in computational design. We will demonstrate this at hand of three application scenarios which also exhibit a relation to algebraic geometry: (i) Rational curves and surfaces with rational offsets possess various applications in Computer-Aided Manufacturing. Their study and design can be based on Laguerre geometry, where they appear as unconstrained rational curves or surfaces in the so-called isotropic model. (ii) The most elegant discrete versions of principal curvature parameterizations of surfaces are objects of sphere geometries and they form the basis for the construction of smooth surfaces from low degree algebraic patches. (iii) The design of various types of circle patterns on surfaces can be effectively based on sphere geometric models. These patterns only exist on those surfaces which carry at least two families of circles. Their complete classification is a problem of algebraic geometry which has been recently solved by R. Krasauskas and M. Skopenkov.
