The Characterization and Detection of Skewed Symmetry

摘要

Skewed symmetry is the type of pattern that emerges when viewing a mirror symmetric, planar shape obliquely. This paper discusses the orthographic case, but also briefly comments on perspective skewing. Orthographically skewed symmetry is characterized by two features which are also present in perfect mirror symmetry and that are preserved under the skewing, i.e., that are invariant under affine transformations. These are parallelism of the chords, the lines joining corresponding points on each side of the symmetric shape are parallel; and collinearity of the midpoints, the points lying in the middle of each of the corresponding point pairs are collinear so that they form a straight symmetry axis. These two constraints (the parallelism constraint and the collinearity constraint) are taken as point of departure and it is shown how they relate to a set of invariants, which skew mirrored point pairs or contour segments should satisfy if they are to make up corresponding point pairs of a skewed symmetry. Although a method based on such invariants is presented, the major outcome is that the previous constraint pair is equivalent to another one, which does not require a priori knowledge of point correspondence.