Continuous symmetry measures: Finding the closest C₂-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 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).