A constructive approach to triangular trigonometric patches

We construct a constrained trivariate extension of the univariate normalized B-basis of the vector space of trigonometric polynomials of arbitrary (finite) order n defined on any compact interval [0,α], where αis a fixed (shape) parameter in (0,π). Our triangular extension is a normalized linearly independent constrained trivariate trigonometric function system of dimension 3n(n+1)+1 that spans the same vector space of functions as the constrained trivariate extension of the canonical basis of truncated Fourier series of order n over [0,α]. Although the explicit general basis transformation is yet unknown, the coincidence of these vector spaces is proved by means of an appropriate equivalence relation. As a possible application of our triangular extension, we introduce the notion of (rational) triangular trigonometric patches of order n and of singularity free parametrization that could be used as control point based modeling tools in CAGD.

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