Representation and comparison of shapes is a problem with many applications in computer vision and imaging, including object recognition and medical diagnosis. We will discuss some constructions from the theory of conformal mapping which provide ways to represent and compare planar shapes. It is a remarkable fact that conformal maps from the unit disk to a planar domain encode the geometry of the domain in useful and tangible ways. Two examples of the relationship between conformal mapping and geometry are provided by the medial axis and the boundary curvature of a planar domain. Both the medial axis and the boundary curvature can be used in applications to compare and describe shapes and both appear clearly in the conformal structure. Here we introduce some results demonstrating how conformal mapping encodes the geometry of a planar domain.
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