Continuous symmetry measures: Finding the closest C-2-symmetric object or closest reflection-symmetric object using unit quaternions

A closed-form solution is provided for the problem. of finding the closest reflection-symmetric object, and its relation to the problems of finding th e closest achiral object, the closest inversion-symmetric object, and the c losest projection plane is discussed. The key to the solution is reducing t his problem to the problem of finding the best (least squares) c(2) rotatio n between two sets of points, in addition to solving the problem of finding the closest C-2-symmetric object. The solution is derived using the quater nion representation of rotation. It is shown that calculation of the best c (2) rotation can be reduced to the diagonalization of the outer product mat rix of the pair (between the two sets). (C) 1999 John Wiley & Sons, Inc.