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

Acta Acad. Paed.. Agriensis, Sectio Mathematicae 28 (2001) 61-113 ON THE DERIVATIVES OF A SPECIAL FAMILY OF B-SPLINE CURVES Miklós Hoffmann (Eger, Hungary) Abstract. This paper is devoted to the geometrical examination of a family of B-spline curves resulted by the modifiaction of one of their knot values. These curves form a surface, the other parameter lines of which are the paths of the points of the original curve at a fixed parameter value. The first and second derivatives of these curves are examined yielding geometrical results concerning their tangent lines and osculating planes. AMS Classification Number: 68U05 1.Introduction B-spline and NURBS curves are well-known and widely used description methods in computer aided geometric design today. The data structure of these curves are very simple, containing control points, knot values and - in terms of NURBS curves - weights. The modification of the control points and the weights has well-known effects on the curves (see e.g.

[9]), while more sophisticated possibilities of curve modification by these data can be found in [1], [3], [4], [8], [10]. The modification of the knot values also affects the shape of the curves, but this effect has been examined only numerically. Some geometrical aspects of the behavior of a B-spline or NURBS curve modifying one of its knot values have been described recently in [5], [6], [7]. The purpose of this paper is to extend these geometrical representations by examining the curves around the parameter value of the modified knot. Definition. The curve s(u) defined by is called B-spline curve of order k (degree k — 1), where N^n) is the I

t

h normalized B-spline basis function, for the evaluation of which the knots UQ,UI, ..., «rj+Jfc are This research was sponsored by the Hungarian Scientific: Research Foundation (OTKA) No. F032679 and FKP No.

0027/2001. n