Methods for computing barycentric coordinates generalized to irregular n-gons and applications of the same

摘要

A method for deriving barycentric coordinates for a point p within an n-sided polygon is provided wherein, for a particular coordinate w_j, corresponding to the vertex q_j, the method embodies a formula which depends only on the edge pq_j, and the two adjacent angles &dgr;_J and &ggr;_j. Similarly, a method is provided for deriving weights w_ij for expressing a vertex q_i in a mesh representation of an object surface in terms of its one-ring neighbors q_j, ∀j&egr;N(i). For a particular vertex q_j, and neighbor vertex q_j, this method embodies a formula which depends only on the edge q_iq_j, and the two adjacent angles &dgr;_j and &ggr;_j. A method of parameterizing a mesh representation of an object surface using the latter formula is also provided. This method begins with the step of computing the weights w_ij in 3D space (in contrast to parameter space) for each of the vertices in the mesh representation. For a vertex i, i&egr;[1 . . . n], the weights w_ij allow the vertex i to be expressed in terms of its one-ring neighbors j&egr;N(i). The method then proceeds to parameterizing the mesh representation responsive to the weights w_ij.