On simultaneous inner and outer approximation of shapes

Abstract

For compact Euclidean bodies P, Q, we define λ(P, Q) to be smallest ratio r/s where r > 0, s > 0 satisfy sQ′ ⊆ P ⊆ rQ″. Here sQ denotes a scaling of Q by factor s, and Q′, Q″ are some translates of Q. This function λ gives us a new distance function between bodies which, unlike previously studied measures, is invariant under affine transformations. If homothetic bodies are identified, the logarithm of this function is a metric. (Two bodies are homothetic if one can be obtained from the other by scaling and translation). For integer &kgr; ≥ 3, define λ(&kgr;) to be the minimum value such that for each convex polygon P there exists a convex &kgr;-gon Q with λ(P, Q) ≤ λ(&kgr;). Among other results, we prove that 2.118… ≤ λ(3) ≤ 2.25 and λ(&kgr;) = 1 + &THgr;(&kgr;-2). We give an &Ogr;(n2 log2 n) time algorithm which for any input convex n-gon P, finds a triangle T that minimizes λ(T, P) among triangles. But in linear time, we can find a triangle t with λ(t, P) ≤ 2.25. Our study is motivated by the attempt to reduce the complexity of the polygon containment problem, and also the motion planning problem. In each case, we describe algorithms which will run faster when certain implicit slackness parameters of the input are bounded away from 1. These algorithms illustrate a new algorithmic paradigm in computational geometry for coping with complexity.