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

Abstract

A method for deriving barycentric coordinates for a point p within an n-sided polygon is provided wherein, for a particular coordinate w, corresponding to the vertex q, the method embodies a formula which depends only on the edge pq, and the two adjacent angles δand γ. Similarly, a method is provided for deriving weights wfor expressing a vertex qin a mesh representation of an object surface in terms of its one-ring neighbors q, ∀jεN(i). For a particular vertex q, and neighbor vertex q, this method embodies a formula which depends only on the edge qq, and the two adjacent angles δand γ. 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 winD space (in contrast to parameter space) for each of the vertices in the mesh representation. For a vertex i, iε[. . . n], the weights wallow the vertex i to be expressed in terms of its one-ring neighbors jεN(i). The method then proceeds to parameterizing the mesh representation responsive to the weights w.