Fast conversion of dynamic B spline curves into a set of power form polynomial curves

Computation of the characteristic points such as inflection points or cusp on a curve is often necessary in CAGD applications. When a curve is represented in a B-spline form, such computations can be made easier once it is transformed in a set of polynomial curves in a power form. Once a curve is represented in a power form, a point evaluation can be also made faster due to Horner's rule even though some issues of stability remains. In addition, the implicitization process of a parametric curve using a resultant usually requires the geometry represented in a power form. Usual practice of the transformation of a B-spline curve into a set of piecewise polynomial curves in a power form is done by either a knot refinement followed by basis conversions, or applying a Taylor expansion on the B-spline curve for each knot span. Presented in this paper is a new algorithm, called direct expansion, for the problem. The algorithm first locates the coefficients of all the linear terms that make up the basis functions in a knot span, and then the algorithm directly obtains the power form representation of basis functions by expanding the summation of products of appropriate linear terms. Then, a polynomial segment of a knot span can be easily obtained by the summation of products of the basis functions within the knot span with corresponding control points. Repeating this operation for each knot span, all of the polynomials of the B-spline curve can be transformed into a power form.