Researches Of Theories And Applications Of Change-able Degree B-splines

B-splines are main tools for today's CAD modeling system. However, since the nonzero polynomials building them all possess the same degree, so that when constructing a curve comprised of polynomial segments of different degrees, the polynomial segments of low degrees should be presented by the same number of control points as the polynomial seg-ments of the highest degree. Thus, redundant data will accrue and the mount of calculation will be increased. Changeable degree B-splines can overcome this shortcoming. They al-low piecewise polynomials of different degrees, have B-spline-like properties and include B-splines. So they are extensions of B-splines. But the progress of their researches is slow in a long time, because the definitions of the basis functions are lack or the results are limited to low degrees.According to the different continuous orders at knots, this paper defines two terms of changeable degree B-spline basis functions with arbitrary degrees. The changeable degree spline spaces respectively spanned by them are different. The detailed work is as follows.1 To meet the demand that any continuous order at the knot between the two neighbor polynomials with degrees m and n is not less than min{m,n} minus the multiplicity of the knot, we defined a term of changeable degree B-spline basis functions by an inte-gral recursion. These basis functions build the unique normalized B-basis for the space spanned by them, so they possess optimal shape preserving property and are the most direct extensions of B-spline basis functions. Their corresponding changeable degree B-spline curves satisfy the properties of convex hull, geometric invariability, variation diminishing and et al. We also give the formulas of knot insertion and recursive evalua-tion of curves and the changeable degree spline space spanned by the basis functions.2 In practice, the above mentioned continuity may be not enough. Thus any continuous order at the knot between the two neighbor polynomials with degrees m and n needs to be not less than min{m,n} if m≠n, and this continuous order is the highest on the premise that the basis functions exist. To meet the need of this continuity, we defined a term of changeable degree B-spline basis functions by a recursive algorithm. These basis functions have positivity, local support property, partition of unity, linear independence and the highest orders of continuity. Their corresponding curves fulfill convex hull prop-erty, geometric invariability property, local control property and so on. Comparing with the results of Sederberg et al, our changeable degree B-spline curves either include their cases of low degrees or have higher orders of continuity than theirs. We also compare these basis functions with the basis functions with optimal shape preserving property, and give their changeable degree spline space.3 The space spanned by the changeable degree B-spline basis functions with optimal shape preserving property is studied. Another truncated-power-basis-like basis of this space is given. This basis is used for the explicit representations of the basis functions of changeable degree B-splines with optimal shape preserving property, so we can see the basis functions and their corresponding space more clearly. This method of explicit representation is unified. It adapts to the basis functions defined by an analogous inte-gral way, also including the changeable degree B-spline basis functions with the highest orders of continuity.