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Complex Barycentric Coordinates with Applications to Planar Shape Deformation
Author(s) -
Weber Ofir,
BenChen Mirela,
Gotsman Craig
Publication year - 2009
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/j.1467-8659.2009.01399.x
Subject(s) - chen , computer science , graphics , citation , library science , computer graphics (images) , information retrieval , artificial intelligence , geology , paleontology
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