Let P and Q be simple polygons with vertex sets {p(1),...,p(n)} and {q
(1),...,q(n)}, respectively. We present an algorithm to construct a pi
ecewise linear homeomorphism between P and Q mapping each vertex p(i)
is an element of P to q(i) is an element of Q by constructing isomorph
ic triangulations of P and Q. These isomorphic triangulations consist
of O(M log n + n log(2) n) triangles where M is the size of the optima
l (minimum size) solution. The algorithm runs in O(M log n + n log(2)
n) time. We also give an O(n + L + k log k) algorithm for constructing
k pairwise disjoint interior paths between k pairs of vertices in a s
imple polygon on n vertices using O(L + k log k) links. The number L i
s the sum of the interior link distances between the k pairs of vertic
es. (C) 1997 Academic Press.