SYMMETRY FROM SHAPE AND SHAPE FROM SYMMETRY

This article discusses the detection and use of symmetry in planar sha pes. The methods are especially useful for industrial workpieces, wher e symmetry is omnipresent. ''Symmetry'' is interpreted in a broad sens e as repeated, coplanar shape fragments. In particular, fragments that are ''similar'' in the mathematical sense are considered symmetric. A s a general tool for the extraction and analysis of symmetries, ''Arc Length Space'' is proposed. In this space symmetries take on a very si mple form: they correspond to straight-line segments, assuming an appr opriate choice is made for the shapes' contour parameterizations. Reas oning about the possible coexistence of symmetries also becomes easier in this space. Only a restricted number of symmetry patterns can be f ormed. By making appropriate choices for the contour parameters, the e ssential properties of Arc Length Space can be inherited for general v iewpoints. Invariance to affine transformations is a key issue. Specif ic results include the (informal) deduction of the five possible symme try patterns within single connected contour segments, the importance of rotational rather than mirror symmetries for deprojection purposes, and relations between simultaneous symmetries and critical contour po ints.