Complex Barycentric Coordinates with Applications to Planar Shape Deformation

j j j k z T z T z = = ∑ where T is a 2D similarity transformation. Proof: Similarity transformations can be represented using a linear polynomial over the complex plane in the following way. If the similarity transformation T consists of rotation by positive angle θ, uniform scale s and a translation t = tx + ity, then according to the rules of complex numbers: ( ) ( ) i T x iy T z se z t z θ + = = + = α +β Where α and β are complex numbers. Since complex barycentric coordinates reproduce linear and constant functions by definition, we have: ( ) ( ) 1 1 1 1 ( ) ( )( ) ( ) ( ) n n n n