Curve L Of A Class Of Problems In Order

Degree elevation of spline curves is an important technique in surface design and computer aided geometric modeling. It is a fundamental tool in CAGD systems and frequently used in geometric design of composite curves, sweeping and skinning surfaces. This paper studies degree elevation from B-spline curves to geometrically continuous spline curves of arbitrary degree.There are three traditional degree elevation algorithms, including Prautzsch algorithm, Cohen algorithm and Piegl&&TiHer algorithm. In this paper, a new method of endpoint interpolating is presented, which improved those algorithms to be useful to all uniform and non-uniform curves.A B-spl ine curve over a given knot vector is completely determined by its control points. By the use of the equivalence between polynomials and symmetric multiaffine mapcs, it is possible to compute B-spJine control points as values of symmetric multiaffine mapes at a sequence of consecutive knots. This result is used to knot insertion and degree elevation, which has been proved very useful.At last, it comes to geometrically continuous spline curves of arbitrary degree. Based on the concept of universal splines , we obtain geometric constructions for the spline control points, for the Bezier points, for knot insertion and degree elevation. The geometric constructions are based on the intersection of osculating flats. The concept of universal splines is defined in such a way that the intersections are guaranteed to exist. As a result of this development , we obtain a generalization of blossom to geometrically continuous spline curves by intersecting osculating flats.