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

70 Imre Juhász is called B-spIine curve of order k (degree k — 1), where JSF* (ti) is the /^

1 normalized B-spline basis function of order /:, for the evaluation of which the knots «0, tii,

..., un+k are necessary. Points d

; are called control points or de Boor points, while the polygon formed by these points is called control polygon. The j^

1 arc. of this curve has the form j S

j (tt) = u £ [?ij

r «j+i), (j = k - l,...,n)

. l=j-k+

1 The modification of knot tt,- effects the arcs sj (w), (j = i — k + 1, t — k | 2,..., i + k — 2). An arbitrarily chosen point of such an arc, corresponding to the parameter value ü £ [uj

, u

} _f_i

) describes the curve i Sj (u, Ui) = di N[

: (u, Ui)

, Ui e «i+i] l=j-k+

1 which we refer to as the path of the point sj («). In [6] we have proved the following theorem. Theorem 1. Paths (

ü

,«,-) and (u, are rational curves in Ui of degree k-z-1, {z= 0,1,.. .,k- 2). The derivative of these paths for k = 4 is studied in [4]. An important corollary of the above theorem, that will be used in this paper as well, is : Corollary 1. Paths of the arcs Sf-^-fi (u. Uj) and s

t

+£—

2 (w,«i) are straight line segments that are parallel to the sides ,1 and d,_i, d,-, respectively

. In this paper we show some interesting properties of B-spline curve modifications obtained by the symmetric translation of knots u

4- and u,+2fc-3-