Determining conformal transformations in Rn from minimal correspondence data

✍ Scribed by Carsten Cibura; Leo Dorst

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 356 KB
• Volume: 34
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1502

✦ Synopsis

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 R n ). 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.