It is shown that the generalized Ball representation for a polynomial curve is much better suited to degree raising and lowering than the B~zier representation. The generalized Ball basis is then extended to polynomial surfaces over a triangle, and recursive algorithms for evaluation and degree raising are given.
generalized Ball basis, deg, ree raising, degree lowering, triangular polynomial patch
In a paper by Ball 1, a basis was introduced for cubic polynomials over a finite interval, and, in a paper by Said 2, this was extended to polynomials of arbitrary odd degree. Said derived a recursive algorithm for evaluating, at any point, a polynomial expressed by this 'generalized Ball basis', and, in a paper by Goodman and Said 3, it was pointed out that this evaluation is considerably more efficient than the de Casteljau algorithm for evaluating a polynomial expressed in the more usual Bezier form. This paper also showed that the generalized Ball form possesses the same 'shapepreserving' properties as the B~zier form. (For details of the B~zier form, see the paper by Boehm et al.4.) Another advantage of the generalized Ball basis is that it renders trivial the reduction of degree from 2m + 1 to 2m, and this paper investigates further degree raising and lowering for generalized Ball curves. Degree raising and lowering are procedures of considerable practical importance. It may be that a designer finds that a curve of given degree does not possess sufficient flexibility for his/her needs, and one way to gain flexibility is to increase the degree of the curve. However, it is in the design of surfaces that degree raising and lowering are most important. Some algorithms produce surfaces from curve input, and it is necessary that the degrees of these curves be modified until they all have the same degree (see the paper by Farin s, p 46). Similarly, in the calculation of the intersection of two surfaces, it may be necessary to ensure that the surfaces have the same degree. In shape-preserving interpolation by surfaces, degrees in certain patches may need to be raised to provide the required shape, and degree modification is then needed