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Journal for Geometry and Graphics 15 (2011), No. 1, 029--043
Copyright Heldermann Verlag 2011

Closed Space Curves Made from Circles on Polyhedra

Clark Kimberling
Dept. of Mathematics, University of Evansville, 1800 Lincoln Avenue, Evansville, IN 47722, U.S.A.

Peter J. C. Moses
Moparmatic Co., 1154 Evesham Road, Astwood Bank - Redditch, Worcesteshire B96 6DT, England

Suppose that P is a polyhedron, all of whose faces are regular polygons such that the incircles of adjoined faces are tangent to each other. Various closed space curves are then determined by linking together portions of the circles. This paper examines such biarc curves, concentrating on those which lie not only on P, but also on a sphere. Thirteen of these are called the regular polyhedral polyarcs: two on a tetrahedron, three on a cube, two on an octahedron, four on a dodecahedron, and two on an icosahedron. More general spherical circle-to-circle curves are also considered.

Keywords: biarc, polyarc, regular polyhedra, sphericon, spherical curve, quadrarc.

MSC: 51M20; 51M04, 51N20

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