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Abstract: This paper presents the matrix representation for the hyperbolic polynomial B-spline basis and the algebraic hyperbolic Bé;zier basis in a recursive way, which are both generated over the space Ω_n=span{sinht, cosht, tⁿ⁻³, …, t, 1} in which n is an arbitrary integer larger than or equal to 3. The conversion matrix from the hyperbolic polynomial B-spline basis of arbitrary order to the algebraic hyperbolic Bé;zier basis of the same order is also given by a recursive approach. As examples, the specific expressions of the matrix representation for the hyperbolic polynomial B-spline basis of order 4 and the algebraic hyperbolic Bé;zier basis of order 4 are given, and we also construct the conversion matrix between the two bases of order 4 by the method proposed in the paper. The results in this paper are useful for the evaluation and conversion of the curves and surfaces constructed by the two bases.

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