Congruence Testing of Point Sets in Three and Four Dimensions

Abstract

I will survey algorithms for testing whether two point sets are congruent, that is, equal upi¾?to an Euclidean isometry. I will introduce the important techniques for congruence testing, namely dimension reduction and pruning, or more generally, condensation. I will illustrate these techniques on the three-dimensional version of the problem, and indicate how they lead for the first time to an algorithm for four dimensions with near-linear running time joint work with Heuna Kim. On the way, we will encounter some beautiful and symmetric mathematical structures, like the regular polytopes, and Hopf-fibrations of the three-dimensional sphere in four dimensions.