Evaluating the Symmetry Transform

Following is a geometric algorithm for deriving the symmetry transform of a shape P having n points with respect to rotational symmetry of order n (Cn-symmetry). This method transforms P into a regular n-gon, keeping the centroid in place as follows  

Minimize over all possible axes of mirror symmetry. 

The minimization performed in step 2 is, in practice, replaced by an analytic solution (derivation and proof can be found in Appendix C). 

We briefly mention the case where the number of points m is less than n, i.e., less than the number of elements in the symmetry group G with respect to which we measure symmetry. In this case, m should be a factor of n such that there exists a subgroup H of G with n/m elements. In this case, we duplicate each point trim times and fold/unfold the points with elements of a left coset of G with respect to H. Following the folding/unfolding method, the relocated points will align on symmetry elements of G (on a reflection plane or on a rotation axis for example). Further details of this case and proof can be found in Ref. 2. 

As described in Section I, the CSM of a set of points with respect to a given symmetry group G is evaluated by first finding the set of points which is G-sym-metric and which is closest to the given set in terms of the average distance squared. We must thus prove that the folding method indeed finds the closest symmetric set of points. The group-theory definitions which are used in this section, are briefly reviewed in Appendix B. 

---

The procedure for evaluating the symmetry transform for mirror symmetry is similar Given a shape represented by m - 2q points and given an initial guess of the symmetry axis, we apply the folding/unfolding method as follows (see Figure 5) ... 

---