Determining conformal transformations in Rn from minimal correspondence data

Abstract

In this paper, we derive a method to determine a conformal transformation in n-dimensional Euclidean space in closed form given exact correspondences between data. We show that a minimal data set needed for correspondence is a localized vector frame and an additional point. In order to determine the conformal transformation, we use the representation of the conformal model of geometric algebra by extended Vahlen matrices— 2 ×2 matrices with entries from Euclidean geometric algebra (the Clifford algebra of ). This reduces the problem on the determination of a Euclidean orthogonal transformation from given vector correspondences, for which solutions are known. We give a closed form solution for the general case of conformal (in contrast, anti-conformal) transformations, which preserve (in contrast, reverse) angles locally, as well as for the important special case when it is known that the conformal transformation is a rigid body motion—also known as a Euclidean transformation—which additionally preserves Euclidean distances. Copyright © 2011 John Wiley & Sons, Ltd.