Computer Atded Geometric Design 8 (1991) 115-121 North-Holland zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Shape preserving properties of the generalised Ball basis T.N.T. Goodman Department of Mathematics and Computer Science, The Unicersity, Dundee, Scotland, UK DDI 4HN H.B. Said School of Mathematical and Computer Sciences, Unicersiti Sains Malaysia, 11800 Penang, Malaysia Received September 1988 Goodman. T.N.T. and H.B. Said, Shape preserving properties of the generalised Ball basis, Computer Aided Geometric Design 8 (1991) 115-121. We show that the generalised Ball basis for odd degree polynomials on a finite interval has a totally positive collocation matrix and thus possesses the same kind of shape preserving properties as the Bernstein basis, though to a lesser degree. This is proved by constructing a comer cutting algorithm for obtaining the Btzier polygon of a polynomial curve from its control polygon with respect to the generalised Ball basis. Keywords. Generalised Ball basis. totally positive, cutting comer, shape preserving. zyxwvutsrqponmlkjihgfedcbaZYXWVU 1. Introduction In [Ball ‘741 a basis was introduced for cubic polynomials over a finite interval which differed from the usually used Bernstein basis. Recently this has been generalised [Said ‘891 to give a basis for polynomials of arbitrary odd degree over a finite interval. One advantage of this ‘generalised Ball basis’ is that there is a recursive algorithm for evaluating the polynomial at any point which is considerably more efficient than the de Casteljau algorithm for the Bernstein form. Another advantage is that the polynomial reduces to lower degree when interior coefficients coincide. Here we show that the generalised Ball basis has a totally positive collocation matrix and thus the shape of a parametrically defined polynomial curve mimics in many ways the shape of its control polygon with respect to this basis. This is shown by constructing a ‘comer cutting algorithm’ for obtaining the Btzier polygon from the control polygon. The existence of such an algorithm also shows that in many ways the Btzier polygon gives a better guide to the shape of the polynomial curve than does the control polygon for the generalised Ball basis. In Section 2 we discuss for a general polynomial basis the relationships between shape preservation, total positivity and corner cutting algorithms. Then in Section 3 we give explicitly the corner cutting algorithm for the generalised Ball basis and briefly discuss the advantages and disadvantages of this basis as compared to the Bernstein basis. 0167-8396/91/SO3.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)