Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)

JUHÁSZ, I., A shape modification of B-spline curves by symmetric translation of two knots

Acta Acad. Paed.. Agriensis, Sectio Mathematicae 28 (2001) 69-113 A SHAPE MODIFICATION OF B-SPLINE CURVES BY SYMMETRIC TRANSLATION OF TWO KNOTS Imre Juhász (Miskolc, Hungary) Abstract. We study the effect of the symmetric translation of knots U{ and U,;_j_2ik-3 on the shape of B-splinc curves. Wc examine when the points of the i -j- k — 2"' arc of the curve move along straight line segment . Quadric and cubic special cases are studied in detail along with the rational case. 1. Introduction B-spline curve and especially its rational counterpart lias become a de facto standard for the description of curves in nowadays CAD systems. A rational Bspline curve is uniquely determined by its degree, control points, weights and knot values. The shape of the curve can continuously be modified by means of its control points, weights and knots. The effect of control points and weights along with their shape modification opportunities are well known. There are numerous publications dealing with different aspects of the topic, cf. [1], [2], [5], [7-9]. For the time being, knot-based shape modifications of B-spline curves are not elaborated, in the related comprehensive books only uniform parametrization is emphasized due to its easy to evaluate formulae, but neither the theoretical nor the application issues of knotbased modifications are described. In this paper we use the usual definition of normalized B-spline functions and curves as follows: Definition 1. The recursive function Nj (it.) given by the equations r

l(

u) = {

1 if u € t

uj j uj +i

) » J I () otherwise Nj N* (u) = U~

U

j AT/"

1 («) + Uj+

k~

U N& (u) is called normalized B-spline basis function of order k (degree k — 1). The numbers Uj < Uj+

1 £ R are called knot values or simply knots, and 0/0=0 by definition. Definition 2. The curve S

(

ÍÍ

) defined by n S (

U) = E N'

C (

U) d

/ ' U 6 i=0