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

64 M. Hoffman Ii The second derivatives of these curves can also be calculated applying the equation (1) iteratively for the basis functions of degree 3. The second derivatives o>

2

S

l ^ J d

2N? 2= d' du l—i — 3 dis where the coefficients are Ö

2Nt. i-3 du

2 d

2N?_

2 = -3 1 1 Wi + l — U +

1 — u 1 Uj + 1 — Ui-2 \ ui + l — Ui1 Ui+1 — Ui Ui+I — Ui-I Ui +

1 — Ui 1 / 1 Wj' + l — u V-i + 1 — U 1 du

2 U

i

+

1 - Uj_2 \ Ui + 1 - Ui- 1 U» + l - Ui U

i +

1 - U,-_1 U

i + 1 - Ui l í 1 Ui + 1 — U U — Uj _ 1 1 - 3Ui

+

2 - Ui- 1 1 w — u fl — «i-1 Uj + i — Ui U

i + 1 — Ui-I Ui + l — Ui U

i

+2 - U 1 d

2Nf-i du

2 —3 • Ui+2 - Ui U

i +1 1 - + i Ui + 2 - Ui U

i +

1 - Ui 1 Ui + i — U U-Ui-i 1 Uj+2 - Ui- 1 V "í + 1 - ui~ 1 wt + l - ui Ui+1 - Ui- 1 Wj

+i - Ui 1 U — Ui Ui+2 — U 1 + Ui+2 - Ui Ui + i - Ui Uj+2 - Ui Ui+i - Ui - 31 1 u - Ui ,, 1 U - Ui 1 <9u

2 —3 Uj+3 - Ui Ui+2 - Ui Ui+i - Ui Ui+

3 - Ui Ui+

2 - Uf Uj+i - U 1 1 U — Ui 1 U — Ui 1 + Uj+3 - Ui Ui+2 — Ui Uj + i - Ui Uj+3 — Ui Uj+

2 - Uj Uj + l - Uj Now the other family of curves, namely the paths will be considered. The first derivative of this family is d Sj dm £ d< l=i3 dNf_ dui where the coefficients are