Constructing Piecewise Linear

Let P = fp1; : : : ; png and Q = fq1; : : : ; qng be two point sets lying in the interior of rectangles in the plane. We show how to construct a piecewise linear homeomorphism of size O(n2) between the rectangles which maps pi to qi for each i. This bound is optimal in the worst case; i.e., there exist point sets for which any piecewise linear homeomorphism has size (n2). Introduction A homeomorphism is a 1-1, onto, continuous map with continuous inverse. Problems of constructing homeomorphisms arise in cartography, animation and computational uid dynamics. A cartographer may wish to merge two similar maps, perhaps slightly distorting one, so that common landmarks coincide. A computer animator may want to transform one shape into the another, while preserving certain features. An aeronautical engineer using computational uid dynamics may need to map a mesh onto the region surrounding the wing of a plane. Each application places di erent requirements on the choice of homeomorphism. Simplicity, robustness, and complexity of the construction, as well as smoothness, angular distortion and ease of modi cation of the resulting homeomorphism, are factors of varying importance in the various algorithms and techniques used in constructing homeomorphisms. In this paper we consider \piecewise linear" homeomorphisms in which two regions are partitioned into corresponding pieces, linear maps are de ned between the pieces, and the linear maps are combined to give a homeomorphism between the original regions. Such \piecewise linear" homeomorphisms are simple to construct and modify and their complexity can be easily quanti ed and measured. Their primary drawback is their lack of smoothness between pieces and the possibility of unnecessarily introducing large angular distortion. Nevertheless, their innate simplicity recommends them for constructing homeomorphisms between complex regions or for constructing homeomorphisms under many constraints. In [3], Saalfeld proposed using piecewise linear homeomorphisms for map con ation, the process of merging cartographic maps. Given two cartographic maps of the same geographic area with identi ed corresponding point landmarks P = fp1; p2; : : : ; png and Q = fq1; q2; : : : ; qng, de ne a homeomorphism between the two which sends pi to qi. If such a homeomorphism does not introduce too much distortion, it will identify each point in one cartographic map with a reasonably close duplicate in the other. Saalfeld constructed his homeomorphism by partitioning the cartographic maps into corresponding triangles, de ning a linear homeomorphism between the triangles, and combining the triangles to give a piecewise linear homeomorphism. The challenging step is partitioning the cartographic maps into corresponding triangles. Ideally, all vertices of the triangles would come from the original point sets P and Q. However, this is not always possible and additional vertices may be required. (See Figure 1 where any triangulation with vertex set P contains triangle p3; p4; p6 while any triangulation with vertex setQ contains triangle q3; q4; q5 and not triangle q3; q4; q6.) By Euler's formula, the number of triangles used in de ning a piecewise linear homeomorphism and hence its \complexity" is a proportional to the number of additional vertices. Finding the minimum number of additional vertices required is an open problem. This paper shows that in the worst case at most O(n2) vertices are required and gives an O(n2) algorithm for constructing a homeomorphism of that size. By a result of Pach, Shahrokhi and Szegedy, these bounds are asymptotically tight [2]. { 2 { p1 q4 q5 p2 q1 q6 p6 p5 q2 q3 p4 p3 Figure 1: Additional vertices are required to construct isomorphic triangulations of P and Q. Formal De nitions and Statement of Goals A homeomorphism h from region Rp to region Rq is piecewise linear if there is some triangulation of Rp such that h is linear on each triangle in the triangulation. Given a piecewise linear homeomorphism h there are many such triangulations of Rp. De ne the size of a piecewise linear homeomorphism h between compact regions Rp and Rq as the fewest number of vertices, edges and triangles among all such triangulations of Rp. If P = fp1; : : : ; png and Q = fq1; : : : ; qng are point sets in Rp and Rq, respectively, we wish to construct the smallest piecewise linear homeomorphism from Rp to Rq which maps pi to qi for each i. Let h be a piecewise linear homeomorphism from Rp to Rq and let p be a triangulation of Rp such that h is linear on each triangle in p. The map h and triangulation p induce a triangulation q on Rq where each vertex, edge and triangle in p maps to a corresponding vertex, edge and triangle of q. Conversely, assume Rp and Rq have isomorphic triangulations p and q where every vertex, edge and triangle, v; e; t 2 p corresponds to a unique vertex, edge and triangle v0; e0; t0 2 q and this correspondence preserves incidence relations. Such isomorphic triangulations of Rp and Rq are called joint triangulations in [3] and compatible triangulations in [1]. Triangulations p and q de ne a piecewise linear homeomorphism h between Rp and Rq as follows. For every vertex v 2 p corresponding to vertex v0 2 q, let h(v) = v0. For every point p 2 Rp lying in triangle v1; v2; v3 of p, express p in barycentric coordinates as p = 1v1 + 2v2 + 3v3 and let h(p) = 1v0 1 + 2v0 2 + 3v0 3. The map h is a piecewise linear homeomorphism from Rp to Rq. In [3], Saalfeld gives a method for constructing isomorphic triangulations between the convex hull of a point set P = fp1; : : : ; png and Q = fq1; : : : ; qng such that pi and qi are corresponding vertices for each i. Of course, no such triangulation is possible if the vertices in clockwise order around the convex hull of P do not correspond to the vertices in clockwise order around the convex hull of Q. Saalfeld's construction uses an exponential number of