Continuous symmetry measures: Finding the closest C2 -symmetric object or closest reflection-symmetric object using unit quaternions.

Abstract

A closed-form solution is provided for the problem of finding the closest reflection-symmetric object, and its relation to the problems of finding the closest achiral object, the closest inversion-symmetric object, and the closest projection plane is discussed. The key to the solution is reducing this problem to the problem of finding the best (least squares) c2 rotation between two sets of points, in addition to solving the problem of finding the closest C2 -symmetric object. The solution is derived using the quaternion representation of rotation. It is shown that calculation of the best c2 rotation can be reduced to the diagonalization of the outer product matrix of the pair (between the two sets). ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 772-780, 1999.