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

66 M. Hoffman Ii Theorem 3. If we consider the surface si(u,ui),u E [wfc-i, '"n+l] > ui £ [Ui-i,Ui

+i

) then the envelope of the family of B-spline curves s

z (v..

«V) is also the envelope of the family of paths s; (ü, U{

) at the points corresponding to u = U{. Proof. It is sufficient to prove, that the two families of curves have points and tangent lines on common at the points corresponding to the parameter value u — U{. If we fix the parameters u = ü and Ui — u

t then a member of both families of curves has been selected. Substituting these parameters to both of the curves the existence of the common point S; (u,iii

) = s

t (

ü

, űi) immediately follows. For the proof of the common tangent lines the first derivatives of these curves will be used. Substituting the parameter u = U{ to the coefficients after some calculations one can receive, that dNt 3 1 dNt a 1 u

i

+1 - Ui dui u=u, 3 du U —11 , Ui + 1 — Ui- 2 Ui +1 — Ui- 1 ' dNU 1 dNf_

2 1 / Ui - Ui- 2 Ui + 2 du, uru, 3 du U=U Ui+1 Ui- 1 V

ui +

1 — ui2 ui+

2 — 1 1 dNt_

x 1 Ui - Ui- 1 dui 3 du U i Ui +

1 - Ui_ 1 ui+2 - Ui- 1 dNt dm du = 0, which yield, that ÖSi (U

, Ui ) du 1 dsi (u, Ui) 3 dm i.e. the curves have also tangent lines on common at the points of the envelope. With the help of the second derivatives of the coefficient functions the oscillating plane of these curves can also be examined. Theorem 4. The osculating planes of the two families of curves s;(u,iij) and Si (u, Ui) coincide at every point of the envelope and this plane is that of the three control points d;_3, d, _

2 d

;

_i for every U{. Proof. The osculating plane is uniquely defined by the first and second derivatives of the curve. Since Theorem 3 holds for the first derivatives it is sufficient to prove that the second derivatives of these curves are also parallel to each other. Using