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

Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) ^"25 129 SHAPE MODIFICATION OF CUBIC B-SPLINE CURVES BY MEANS OF KNOT PAIRS Róbert Tornai (Debrecen, Hungary) Abstract. The effect of the modification of not consecutive knot values on the shape of B-spline curves is examined in this paper. It is known that an envelope of the one-parameter family of B-spline curves of order k, obtained by the modification of a knot, 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. An extension of shape modification methods are provided for cubic B-spline curves, that utilize this envelope. This paper extends the possibilities for choosing the new position of a point of the curve by allowing to modify knots that are not consecutive. A MS Classification Number: 68U05 1. Introduction Computer aided design widely use B-spline curves and their rational generalizations (NURBS curves) that play central role today. Besides, they are used in computer graphics and animations. These curves are excellent tools in design systems to create new objects, but the modification and shape control of the existing objects are also essential. The data structure of a B-spline curve of order k is fairly simple. It only consists of control points and knot values. Hence shape control met hods can modify such curves only by altering these data. One of the most comprehensive books of this field is [9] where shape modifications, based on control point repositioning are also described. Some publications discuss shape modifications, e.g., [10] which present constraint-based curve manipulations of curves of arbitrary degree and basis functions. [11] proposes direct modification of free-form curves by displacement functions, which method comprises knot refinement and removal, control point repositioning and degree elevation. Some aspects of knot modification is also been studied, like in [12] where the effect of knot variation is examined from numerical point of view. Several papers and articles investigate the choice of knot values in curve approximation and interpolation, cf. the recently published [13] and the references therein. It is an obvious fact, that the modification of the knot vector affects the shape of the curve. Some results concerning the geometric aspects of knot modifications have already been presented by Juhász and Hoffmann for B-spline curves in [2],