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

76 Imre Juhász specified by control points d

0, di, . .., d

n

, weights Wo, ivi, . . ., w

n (w{ > 0) and knots (uo) Ui, . . ., tí

n+fc), (el. = 2,3), can be produced by projecting the integral B-spline curve, determined by the same knots and the control points Wo do u;idi w

nd

n wo w

n m R

d+

1, from the origin onto the hyperplane w = 1. Thus rational B-spline curves inherit those properties of integral B-spline curves that are invariant under central projection. Therefore Theorem 2 is valid for rational B-spline curves either. Properties of the case k — 3 remains valid, since central projection preserves incidence and straight lines. The case k = 4 changes in part, since central projection does not preserve parallelism. For this reason, points of the arc Sj-j-2 (u) move along straight line segments, the extension of which intersect the side d;, d,;

+

1 at its inner points moreover, intersect the line determined by the point of homogeneous coordinates (u>;_idj_i — wid

?

, Wi-i — W{) and 3d;

+2 — it>i+idj_|_i, Wi+2 ~ t^i+i)- (This line is the vanishing line of the plane direction of Subsection 2.2 in this central projection.) 4. Conclusions In this paper we have examined the shape modification effect of the symmetric translation of knots U{ and 2jt-3- We proved that this symmetric translation moves points of the arc s

l

+k-2 (u) along straight line segments, if and only if, Ui+k-i — Ui = Ui+2k-3 — ui+k.-2 holds. We studied the k — 3 and k = 4 special cases in detail, and carried over the results to rational B-spline curves as well. Further research is needed 011 the simultaneous modification of knots to reveal their shape modification possibilities. References [1] Au, C. K., YUEN, M. M. F., Unified approach to NURBS curve shape modification, Com-puter-Aided Design

, 27 (1995), 85-93. [2] FOWLER, B., BARTELS, R., Constraint-based curve manipulation, IEEE Computer Graphics and Applications

, 13 (1993), 43-49. [3] HOFFMANN, M., JUHÁSZ

, I., Shape Control of Cubic B-spline and NURBS Curves by Knot Modifications, in Banissi, E., Khosrowshahi, F., Srafraz, M., Ursyn, A. (Eels.) Proceedings of the Fifth International Conference on Information Visualisation, 25-27 July 2001, London, England, IEEE Computer Society, 63-68.