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 65 dNt

z _ n

i

+

1 - u u

1 + i - u Uj+i - u dui Ui +

1 — Ui-2 Ui+1 — t-'i-l (tl

i +

! — U;) dNi_ 2 u ~ ui2 Ui

+1 — U Ui +

1 — U dui Ui+

1 — 2 Ui + 1 — ui- 1 (U{ +

1 — Ui)' + H+2 U — Ui1 u»+i - u + «£+2 — \ Wí + 1 ~ «i-1 (tij + i — Ui)" lt

l

+2 — u u — Ui ^ Ui+

2 — u u — ui+l (u

i

+

2 - Ui)" «i+1 - Ui U

i

+

2 - Ui (u

l + l - Ui)' dN?_

x _ U-UjI f U- Uj-I Uj+X- U dlli U

i

+

2 - Ui- 1 I U

i + 1 - Ui1 (Ui + l - Ui) Ui + 2 — U U — Ui Ui+2 — U U — Ui +

1 + (ui

+

2 - «»+1 - Ui lli+

2 - Ui (Ui+i - Ui)' Ui+ 3 — ti ii — Ui U — Ui («1+3 — Ui)" u?:+2 — Ui+i ~ ui Ui+ 3 - u / I/. - tt,+

2 « - »

j U - U,- U - lij+1 + ~ — ~r Ui+

3 — Uj \ (Uj + 2 — Ui)" Ui + i — Ui Ui+2 ~ Ui (Ui + i ~ Ui dNf _ U- Uj+

3 U - Ui U - Ui Ölti (

Ui+

3 _ U

if Ui+2 - Ui Ui+i Ui U — Ui / U — Uj+2 U - Ui U — Ui U — Ui \ -j- I _ -J- _ I _ Ui+3 — Ui y(Ui+

2 — Ui)" Ui + l — Ui Ui+2 — Ui (ui+i — Ui)" J The second dertivatives of these paths are the following <s>

2Sj _ Á FN? Ou} - 1 du? • where the coefficient functions are large polynomials thus, for the sake of brevity they are not presented here. 2. New results Using the derivatives of the preceeding section the following theorems can be proved (in these proofs the Maple software was applied for the evaluation and simplification of polynomials):