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On Evolute Cusps and Skeleton Bifurcations Alexander Belyaev And On Evolute Cusps and Skeleton Bifurcations Alexander Belyaev and Shin Yoshizawa The University of Aizu Aizu-Wakamatsu 965-8580 Japan g fbelyaev, m5031126 @u-aizu.ac.jp Abstract Fig. 2 demonstrates the description of the skeleton of a figure as the set of centers of maximal disks inscribed in the Consider a 2D smooth closed curve evolving in time, the figure. skeleton (medial axis) of the figure bounded by the curve, and the evolute of the curve. A new branch of the skeleton can appear / disappear when an evolute cusp intersects the skeleton. In this paper, we describe exact conditions of the skeleton bifurcations corresponding to such intersections. Similar results are also obtained for 3D surfaces evolving in time. Figure 2. Left: a \bison" gure, its skeleton, and a maximal disk inscrib ed in the gure; the b old tisapoint of tangency b etw een the disk and Introduction poin the b oundary of the gure. Right: the skeleton of a gure can b e de ned as the set of centers of The skeleton or medial axis of a planar figure is the clo- maximal disks inscrib ed in the gure. sure of the set of centers of maximal disks contained inside the figure, i.e., those discs contained in the figure but in no other disc in the figure. The skeleton is an important shape Consider a figure evolving in time. How does the skele- descriptor invented by Blum [6]. It is extensively used in ton of the figure change? An analytical description of how connection with human shape perception theories [5], [9]. smooth points of the skeleton evolve under a general bound- Fig. 1 shows simple closed curves mimicking silhouettes of ary evolution is given in [1] for the 2D case. This paper ad- a bison, a fish, and a human and their skeletons. dresses conditions of topological changes of the skeleton. Fig. 3 demonstrates two topological changes of the skeleton of an evolving curve: in the “head” and in the “chest”. In this paper, we deal with the skeleton bifurcations similar to the bifurcation happened in the “head” where a new skele- ton branch has grown from an inner point of the skeleton. Let us recall that the best approximation of a curve at its point P by a circle is given by the circle of curvature called P also the osculating circle at P [12]. If is an inflection point (where the curvature is zero), the circle of curvature degenerates into the tangent line at P . The locus of the centers of curvature of a given curve is called the evolute of the curve. The evolute of a smooth curve has singularities (so-called cusps). A rigorous mathematical description of the relations be- es mimicking silhou- Figure 1. Simple closed curv tween the skeleton and the evolute (the focal surface in 3D) of a bison, a sh, and a h uman and their ettes of a simple closed curve (a simple closed surface in 3D) eletons. sk evolving in time constitutes the main results of the paper. The 2D version of the skeleton branching theorem de- Proceedings of the International Conference on Shape Modeling and Applications (SMI ’01) 0-7695-0853-7/01 $10.00 © 2001 IEEE Similarly, 0 n = k t; 00 0 0 2 n =(k t) = k t k n; 00 0 00 3 0 n =(k + k ) t 3k k n; n(s + )=n(s)+ 2 3 0 3 00 4 k k + t k + k + O ( ) + 2 6 3 2 2 0 4 k kk + O ( ) ; + n 2 2 2 00 0 3 k k (s + )=k (s)+ k + O ( + ); Figure 3. Two di erentskeleton bifurcations hap- 2 pened in the \head" and in the \c hest". In the 0 0 2 00 1 k k k 1 \head" the new skeleton branch has grown from 2 3 = + + O ( ): 2 3 2 an inner p ointoftheskeleton. k (s + ) k (s) k k 2k (s + ) For the evolute point e we have (s + ) scribed in the next section was discovered while playing n (s + )=r(s + )+ = e(s)+ 1 e (1) (s + ) with the Java 2D Curve Simulator described in [14]. k 0 0 2 00 k k k 2 3 4 + t + + O ( ) + 2 k 3k 2k 2D Skeleton Branching Theorem 2 00 0 2 0 k k k 3 2 + O ( + ) + n r = r(s) 3 2 Let us consider a planar curve parameterized 2 k k 2k by arc length s. To study the shape of the evolute of the 0 r(s) r(s + ) (s) 6=0 curve at a small vicinity of a point we expand Thus, if k , in a small vicinity of the center of cur- 1 (s)+n(s)=k (s) into Taylor series with respect to , . Let vature r the evolute is approximated by a (s) parabola tangent to the normal n (see the left image of 0 ? r = t and n = t 0 k s)=0 Fig. 4). If ( then (1) simplifies into = r(s) compose the Frenet frame at r . The evolute is given (s + ) e(s)= e (2) by 00 00 k k n(s) 4 3 3 2 + O ( + O ( = t ) ) + n : e(s)=r(s)+ ; 2 3k 2k k (s) 0 00 (s) r(s) where k is the curvature of the curve at . According k For a generic curve k and do not vanish simultaneously. to the Frenet formulas Thus, in a small vicinity of the center of curvature, the evo- 3 2 y = Ax 0 0 lute is approximated by a semicubic parabola t = k n; n = k t (s) tangent to the normal n . See the middle and right im- we have ages of Fig. 4. 0 00 0 000 0 0 2 r = t; r = t = k n; r =(k n) = k n k : t evolute center of center of curvature It yields curvature evolute center of evolute 3 2 curvature 00 00 0 0 4 r r r(s + ) = r + r + + O + = n n n 2 6 3 t t t 2 4 k + O + = r(s)+t 6 Figure 4. 3 2 0 4 k + k + O + n : 6 2 A vertex of a smooth curve is a point of the curve where 1 0 =0 http://www.u-aizu.ac.jp/ m5031126/Research/CCurve/CCurve.html the curvature takes an extremal value, k . Proceedings of the International Conference on Shape Modeling and Applications (SMI ’01) 0-7695-0853-7/01 $10.00 © 2001 IEEE (t) Proposition 1 The evolute of a curve r has a cusp at t r(t ) 0 0 if and only if is a vertex of the curve. The cusp on the evolute is pointing towards or away from the vertex according as the absolute value of the curvature has a local t minimum or maximum at 0 . Now the proposition follows immediately from (2). A dif- ferent proof can be found in [10]. For a generic curve, the osculating circles centered at or- Figure 5. Left: A closed curv e; its sk eleton and dinary evolute points intersect the curve. The osculating ev olute; an osculating circle centered at a cusp of circles centered at the cusps of the evolute have either inner the evolute and its radial segment connecting the or outer contacts with the curve. The previous proposition cusp and the tangency p oint; an inner bitangent can be reformulated as follows. circle touching the curve at the same p oint as the tained by Proposition 2 Given a generic curve, the osculating circle osculating circle do es and, therefore, con Right: the curve is mo di ed centered at an evolute cusp pointing towards (away from) the osculating circle. h that the radial segment connecting the cusp the curve has an inner (outer) contact with the curve. suc and the tangency point intersects the sk eleton; eleton branchended The circle of curvature centered at a skeleton endpoint note app earance of a new sk can be considered as a limit of the inner bitanget circles at the cusp. (i.e., touching the curve at least twice) whose centers form the skeleton. Thus the circle is osculating and it has an inner contact with the boundary of the figure. If the boundary of to see that the function has two equal absolute minima if the figure is oriented by its inner normal, the endpoints of and only if Q is a generic point of the skeleton of the figure the skeleton correspond to positive maxima of the curvature bounded by the curve. Let us move Q along the skeleton to- of the boundary and we arrive at the following result. wards a skeleton endpoint. A degenerate absolute minimum Proposition 3 The endpoints of the skeleton of a figure are appears when Q reaches the endpoint and the absolute min- located at cusps of the evolute of the boundary of the fig- ima merge. See Fig. 6. An inner circle has a high-order ure. Those cusps are pointing towards the boundary of the contact with the curve if and only if the distance-squared figure. function with Q located at the center of the circle has a de- generate absolute minimum. This proves the proposition. However the evolute cusps do not necessary coincide with the endpoints of the skeleton of the figure bounded by degenerate the curve. If the osculating circle centered at a cusp of the critical point evolute does not lie inside the figure, then the cusp is not a skeleton endpoint.
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