A New Exact Evaluation Formula for Loop Subdivision Surfaces

A block diagonal matrix can be obtained by utilizing the discrete Fourier transform （DFT） and the symmetrical structure called 1-ring and 2-ring in the vicinity of an extraordinary point on a mesh.By using in a sequence the eigen-decomposition,a permutation,the inverse discrete Fourier transform （IDFT） and a truncation on the matrix,an exact and explicit parametrization formula for Loop subdivision surfaces is proposed.