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

HOFFMANN, M., On the derivatives of a special family of B-spline curves

62 M. Hoffman Ii necessary. The points d; are called control points or de Boor-points, while the polygon formed by these points is called control polygon. Definition. The j

t

h span of the B-spline curve can be written as j sj (u) = y diN

{

k (u) , u G [uj,u

j+

1) . i-j-k+i Modifying the knot -Uj, the point of this span associated with the fixed parameter value ü G [iij,Uj+1) will move along the curve j Sj (v, Ui

) = Nf (Ű, Ui

) d/, Ui G [ui_x, Ui+i] • l=j-k+

1 Hereafter, we refer to this curve as the path of the point sj (u). In [5] and [6] Juhász and Hoffmann proved important properties of these paths, among which the most important is the following Theorem 1. Modifying the knot value Ui E ['"2-1, ti;+i] of the k

t

h order B-spline curve, the points of the spans ..., Sj+&_2(ii) moves along rational curves. The degree of these paths decreases symmetrically from k — 1 to 1 as the indices of the spans getting farther from i, i.e. the paths s,:_

m(ii, ni) and Sj_|_

m_i(it, U{) rational curves of degree k — to with respect to Ui, (m = 1,

..., k — 1). Beside these paths we can also consider the one-parameter family of B-spline curves n s (u, űi) = E d'' N'

k (

u' ) 1 u e C

Ujf

c 1' U

n

+

1 ] / = 0 yielded by the modification of the knot value u;. In trems of these curves another property has been proved by Juhász and Hoffmann (see [6]), namely the family of these curves has an envelope, which is also a B-spline curve. Theorem 2. The family of the k

t

h order B-spline curves s(u,ui), (k > 2) has an envelope. The envelope is also a B-spline curve of order (k — 1) and can be written in the form i-l b (u) = V d,N

k1 (u) , v G [vi-i, in]

, l=i-k + l where Vj = Uj if j < i and Vj = Uj+i otherwise, that is the i

i

h knot value is removed from the knot vector uj of the original curves.