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

OIL the derivatives of a special family of B-spline curves 67 the second derivatives of the coefficient functions and substituting the parameter value u = u-i the following result can be obtained: d

2Nt_ 3 1 d

2Nt_ dui

2 ll=Ui 3 du 2 d

2N?_2 1 d

2Nf_ dui

2 U =11 , 3 du 2 d

2Nl 1 1 d

2 N?_ dúr u=u

, 3 du 2 d

2Nt d

2Nf dm

2 u=u

, du

2 U= 21 = 2 = 2 Ui-l-l — 'Ui-2 Wj + 1 — V-i- 1 — Uj+

i + Ui-2 - Ui12 + Ui- 1 ~U

i

+

2 + Ui-1) (Ui+i - Ui-1) ( —«»+1 + Ui2) ' 1 u =u

, = 0, (Ui +

1 - Uf_i) (Uj+ 2 - Uf_i) ' which immediately yield, that 8

2Si (u, Ui] du

2 1 d

2s i(u,ui) 3 <9ur Hence the osculating planes of the two families of curves coincide at the parameter values u — Uj. Moreover, the second derivatives do no depend on Uj, and using the notations d

2NU A :-d

2Nt_ 3 duj

2 B := <9u

?2 they can be written in the form c?

2s

)

: ( ti, t/i) dm

2 d

2Si (u, Ui) du

2 — A (d,;_3 dj — o) "f" B ( di —

2 d;

1) , •I A (di_3 - d,;_

2) + iß (dí_2 - dj -1) . This means that these derivative vectors are in the plane of the control points d,_3, d,_2, d;_i for every Ui

. The same holds for the first derivative vectors since the envelope is a quadratic B-spline curve (a parabola) defined by these control points and it has common tangent lines with both of the families of the curves at u — Uj. This yields, that the osculating planes of the curves coincide with the plane of the three control points mentioned above for every Ui. 4. Further Research Some geometrical aspects of the modification of a knot value of a cubic B-spline curve have been discussed. Defining a special surface with two families of curves it turned out that these two families have the same envelope at a certain parameter