Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2004. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 31)

TORNAI , R. , Shape modification of cubic B-spline curves by means of knot pairs

62 R. Tornai [3] and [6], where the main result was the following: the one-parameter family of B-spline curves of order k, resulted by the modification of a knot, possesses an envelope which is also a B-spline curve of the same control polygon and of order k — m, where m is the multiplicity of the modified knot. This envelope can be used for geometric constraint-based shape modification of cubic B-spline curves. This property forms the basis of constrained modification of the curve which first outlined in [1] and discussed in a detailed form in [4] and [7]. Further special shape control techniques discussed in [8]. In terms of surfaces t he theoretical generalization of these theorems can be found in [5]. In this paper I extend the possibilities of a shape control method described in [4] and [7] by letting not necessarily neighboring knots to change. 2. Modifying a knot Definition 1. The curve s (u

) defined by n s (

u) = N[k d

/ ' " G i

Uk-h wn+l] 1=0 is called B-spline curve of order k (degree k — 1), (1 < A: < n + 1), where (u

) is the Ith normalized B-spline basis function of order k

, for the evaluation of which the knots uo, ui,

..., u

n

+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 jth arc of the B-spline curve of Definition 1 is of the form j *j («) = dl

Ni

k (

u)' u G ' 0' = * - 1) • • • * n) i=j-k+1 The modification of the knot value u,- alters the shape of the arcs sj (u), j = i — k + 1, i — k + 2,..., i + k — 2. The point of such an arc that corresponds to an arbitrarily chosen parameter value ü £ describes the curve j Sj (Ű, Uj) = ("> Ui) > Ui £ [«*-1) Ui+1] • l=j-k +

1 In [2] Juhász and Hoffmann proved the following property. Theorem 1. Altering a knot value uG [wi-i, w»+i) of a B-spline curve s(u) of order k (k > 2), the one-parameter family of B-spline curves n s(u,uj) = y^ diN

t

h («, Ui) , U e [iifc-l, Mn+l] /=0