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# v-kfdp-89 - Kink-free deformations of polygons Gert Vegter...

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Kink-free deformations of polygons Gert Vegter Dept. of Computing Science, University of Groningen, P.O.Box 800, 9700AV Groningen, The Netherlands Abstract We consider a discrete version of the Whitney- Graustein theorem concerning regular equivalence of closed curves. Two regular polygons P and P’, i.e. polygons without overlapping adjacent edges, are called regularly equivalent if there is a continuous one-parameter family Pa, 0 5 s 5 1, of regular poly- gons with PO = P and PI = P’. Geometrically the one-parameter family is a kink-free deformation transforming P into P’. The winding number of a polygon is a complete invariant of its regular equiva- lence class. We develop a linear algorithm that deter- mines a linear number of elementary steps to deform a regular polygon into any other regular polygon with the same winding number. 1 Introduction The Whitney-Graustein theorem states that up to regular (‘kink-free’) deformation a regular closed curve in the plane is completely determined by its winding number. In this paper we consider the dis- crete version of this theorem in which closed curves are replaced with polygons. A setting for this prob- lem is given in Section 2. In Section 3 we show how to reduce a polygon to isothetic form in which every edge is parallel to one of the coordinate axes. Sec- tion 4 contains the algorithm that reduces an isothetic polygon to isothetic normal form that is uniquely de- termined by its winding number. In Section 5 we Permission to copy without fee a11 or part of this material is granted provided that the copies are not made or distributed for direct com- mercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 0 1989 ACM 0-89791-318-3/89/0006/0061 \$1.50 indicate possible extensions of this work to the clas- sification of polygonal curves on piecewise flat sur- faces. This paper was inspired by similar work of Mehlhorn and Yap, cf. [MY]. They derive an algo- rithm that computes a quadratic number of elemen- tary steps to transform between regularly equivalent polygons. The sequence of steps can be determined in linear time. Their normal form as well as their method is however completely different from ours. Acknowledgements The author is grateful to Chee Yap for communicating the work [MY] to him, and to Jan Tijmen Udding for helpful criticism. 2 Equivalence of Regular Poly- gons A closed curve C in the plane is called regular if it has a smooth parametrization f : [0, l] -+ R2 such that f(0) = f(l), S(O) = f’(1) and f’(t) # 0 for 0 5 t 2 1. The winding number w(C) of C is the degree of the map f* : S1 + S1 defined by f+(t) = f’(t)/lf’(t)l- The regularity of f allows us to consider t as a pa- rameter on the unit-circle 9. Geometrically speaking the winding number of C is the number of full turns - counted with sign - of the tangent of C as we go around C. The figure-of-eight has winding number 0,

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