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 63 Hence two families of curves have been received, the paths of the points and the B-spline curves themselves. These two families of curves can be considered as parameterlines of the surface patch n S (it, lli) = ( U> U*') ' U G [ ufc-l> Un + l] , Ui G [«»-1, «i+l) • / = 0 The envelope mentioned above in Theorem 2. is a curve on this surface, but the parameter lines behave in a singular way at the points of that curve. We have seen that it is an envelope of the family of B-spline curves. In the next sections, where we will restrict our consideration to the cubic case (k = 4) the derivatives of the two families of curves will be computed in the points of the quadratic envelope by the help of which we will prove, that this curve is also the envelope of the paths and both families have the same osculating plane at every point of this envelope, which plane is also the plane of the envelope itself. 2. The derivatives of the curves Let the knot value U{ of a cubic B-spline curve defined above be modified. At first the family of B-spline curves will be considered, the derivatives of which can be calculated by a well-known iterative formula, which can be found e.g. in [9]: (1) ^ = E d' 3i— N?(u,ui) 1 N? + I(u tui)) Ő U \Ul + 3-Ul U/+4-U/+1 + J Using this rule the first derivatives of the coefficients are <9^3 __ 3 1 «i+i - « «i+l - « dNf_ 2 du dNU du =3 II j +1 — U{ _ 2 Ui +1 — Ii»- 1 ui + 1 — ui 1 U i +1 - U llj + j - U «i + l — lit — 2 «i + l — «i-1 «t + 1 — Iii 1 Ii - «i-1 «i + l - Ii «j+2 - U Ii - Ui «i+2 - «i-l \«i+l - «i-1 «i + l - Ui «i+2 - «i «i + l - «i 1 « - «i_i «i + i - « ^ «i+2 - Ii U - Ui «i+2 - «i-l \«i + l - «i-l «i+l - «i «i+2 - «i «i + l - «i 1 « — Ui U — Ui ön: =3 «i+3 — «i «i+2 — «t «i + l — «i 1 U — Iii U — Ui '« «i+3 - «i «i+2 - «i «i + l - «i
