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Unformatted text preview: SIAM J.COMPUT. Vol. 20, No. 4, pp. 603621, August 1991 1991 Society for Industrial and Applied Mathematics 001 CONSTRUCTIVE WHITNEYGRAUSTEIN THEOREM: OR H O W TO UNTANGLE CLOSED PLANAR CURVES* KURT MEHLHORN A N D CHEEKENG YAP Abstract. The classification of polygons is considered in which two polygons are regularly equivalent if one can be continuously transformed into the other such that for each intermediate polygon, no two adjacentedgesoverlap.A discrete analogue oftheclassicWhitneyGrausteintheorem isprovenby showing that the winding number ofpolygons isa complete invariant for this classification. Moreover,this proofis constructiveinthatforanypairofequivalentpolygons,itproducessome sequenceof regular transformations taking one polygon to the other. Although this sequence has a quadratic number oftransformations, itcan be described and computed in real time. Key words, polygons, computational algebraictopology, computational geometry, WhitneyGraustein theorem, winding number AMS(MOS) subject classifications. 68Q20, 55M25 1. Why a circle differs from a figureofeight. First consider closed planar curves that are smooth. Intuitively, a "kink" on such a curve is a point without a unique tangentline.Itseems obvious thatthereisno continuous deformation offigureofeight to a circle in which allthe intermediate curves remain kinkfree (see Fig. 1). Figure 2 shows another curve thatclearlyhas a kinkfree deformation to a circle. Letus make thisprecise.Bya (closedplanar)curvewe mean acontinuous function C :[0,1]>E 2, C(0)= C(1),where E 2 isthe Euclidean plane. The curve C is regular if the first derivative C(t) is defined and not equal to zero for all t[0, 1], and C(0) C(1).Let h:[0,1 [0, 1> E 2 be a homotopy between curves Co and C1, i.e., h is a continuous function and each Cs :[0,1]>E 2 (0< s  < 1)is a curve, where we define Cs(t)=h(s,t)(t[0, 1]).The homotopy is regular ifeach C is regular. Two regular curves are regularly equivalent ifthere is a regular homotopy between them. A classical result known as the WhitneyGraustein theorem [5], [1] says that two curves are regularly equivalent ifand only ifthey have the same winding number (up FIG. 1. Transforming afigureofeightto a circle" kink appears. *ReceivedbytheeditorsFebruary3,1988;acceptedforpublication(inrevised form)September12,1990. ?Informatik, Universitaet des Saarlandes, D6600 Saarbruecken, Federal Republic of Germany. The work ofthis author was supported by Deutsche Forschungsgemeinschaft grant Me61. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012. The work ofthis author was supported by National Science Foundation grants DCR84 01898 and DCR8401633....
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