As a measure for the resemblance of curves in arbitrary dimensions we consider the so-called Fréchet-distance, which is compatible with parametrizations of the curves. For polygonal chains P and Q consisting of p and q edges an algorithm of runtime O(pq log(pq)) measuring the Fréchet-distance between P and Q is developed. Then some important variants are considered, namely the Fréchet-distance for closed curves, the nonmonotone Fréchet-distance and a distance function derived from the Fréchet-distance measuring whether P resembles some part of the curve Q.

Abstract.The problem of minimizing /[Vu|2 + q2(x)\2(v)] dx in an appropriate class of functions v is considered. Here q(x) ¥= 0 and A2(t>) = X2 if v < 0, = X22 if v > 0. Any minimizer u is harmonic in {u ¥= 0} and | Vu|2 has a jump a2(x)(\]-\22) across the free boundary [u ¥= 0). Regularity and various properties are established for the minimizer u and for the free boundary.

No abstract

Abstract.The problem of minimizing /[Vu|2 + q2(x)\2(v)] dx in an appropriate class of functions v is considered. Here q(x) ¥= 0 and A2(t>) = X2 if v < 0, = X22 if v > 0. Any minimizer u is harmonic in {u ¥= 0} and | Vu|2 has a jump a2(x)(\]-\22) across the free boundary [u ¥= 0). Regularity and various properties are established for the minimizer u and for the free boundary.

We consider the problem of computing geometric transformations (rotation, translation, reflexion) that map a point set A exactly or approximately into a point set B. We derive efficient algorithms for various cases (Euclidean or maximum metric, translation or rotation, or general congruence).