A matrix Method for Computing the Derivatives of Interval Uniform B-Spline Curves

The matrix forms for curves and surfaces were largely promoted in CAD. These formulations are very compact to write, simple to program, and clear to understand. They manifest the desired basis as a matrix transformation of the common power basis. Furthermore, this implementation can be made extremely fast if appropriate matrix facilities are available in either hardware or software. Derivatives are very important in computation in engineering practice on graphics structures. B-spline functions are defined recursive, so direct computation is very difficult. A method for obtaining the matrix representations of uniform B-splines and Bezier curves of arbitrary degrees have been presented in this paper. By means of the basis matrix, the matrix representations of uniform B-splines and Bezier curves are unified by a recursive formula. The four fixed uniform Kharitonov's polynomials (four fixed uniform B-spline curves) associated with the original interval uniform B-spline curve are obtained in matrix form. The fixed control points of the derivatives of the four fixed uniform Kharitonov's polynomials (four fixed uniform Bspline curves) are found. Finally the interval control points of the derivative of the interval B-spline curve is computed from the fixed control points of the derivatives of the four fixed uniform Kharitonov's polynomials (four fixed uniform Bspline curves). A numerical example is included in order to demonstrate the effectiveness of the proposed method. Index Term— Recursive matrix representations, interval B-spline curve, CAD, derivatives of B-spline curve, CAGD.