Matrix Representation And Conversion Matrix For Two Bases Of The Algebraic Hyperbolic Space

The limitations caused by the rational model in NURBS (Non-Uniform Rational B-Spline) curves and surfaces motivate the broad research of new representation of curves and surfaces. Among that research, the uniform hyperbolic polynomial (HP) B-Spline basis and the algebraic hyperbolic (AH) Bezier basis which are both defined in the algebraic space H_n={sinht, cosht, t^n-3,..., t,1}(n≥3) have been largely studied for their similar properties to the uniform B-spline basis and the Bernstein basis respectively.This paper presents the matrix representation for the HP B-Spline basis in a recursive way, which is generated over the algebraic space H_n. This definition of the matrix form is more straightforward and legible than the definition in the recursive and integral approach. As examples, the specific expressions of the matrix representation for HP B-Spline basis of order 4 are given by the recursive method.Similarly, the matrix representation for the AH Bezier basis over H_n. is derived in the recursive approach too. At the same time, this paper also gets the specific expressions of the matrix representation for AH Bezier basis of order 4 by employing the recursive results.At last, the conversion matrix from the AH Bezier basis to the HP B-Spline basis of the same order is also given by a recursive approach. The conversion matrix between both bases not only has the practical value, but also proposes a new tool for the theoretic research about the two bases. We also construct the conversion matrix between the two bases of order 4 by the method proposed in this paper. These results constitute the primary contents of this paper.As we know, it is both convenient and practical to describe curves and surfaces by matrix representation in CAGD, such as facilitating the evaluation and conversion of the curves and surfaces. The matrix forms for curves and surfaces are largely promoted in CAD. So we expect the results concerning the matrix form of the HP B-Spline basis and the AH Bezier basis on the algebraic space H_n can be employed in future CAD/CAM systems.