Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)
JUHÁSZ, I., A shape modification of B-spline curves by symmetric translation of two knots
A shape modification of B-spline curves by symmetric translation of two knots 71 2. Symmetric translation of knots Ui and Ui+2k-3 We study how points of a B-spline curve move, i.e. what will the paths be like, when knots Ui and Ui+ 2k-3 are symmetrically translated. By symmetric translation of knots Ui and Uj , (i < j ) we mean the u x + A, Uj — A, A G R type modification. In order to preserve the monotony of knot values A can not take any value but it has to be within the range [—c, c], c = min{uj — Wj+i — Ui, uj — Uj-i, Uj+i — Uj}. Under the circumstances, the i + k — arc of the B-spline curve, that is effected by both Ui and Ui+2k-3, has the form i+k-2 l=i1 i+k4 Ui+k-U A Tk — l l=i+ 1 = V d,N? („) + [N*1 (») + <V («) d, (1) \ ui+2fc-4 - J U i+ k- 1 - U jfc.i H ^ («) (d,-_i - di) Ui+k-l — «i + "~ "' T (u) ( d' + t2 - di+ i3 ) • Further on we examine the B-spline curves Si+k~ 2(u,\) and their paths obtained by the substitution = u» + A and w»+2jfc-3 = «;+2ifc-3 — A. Theorem 2. Paths Si+k2 (u, A) , A G [—c, c] are straight line segments, if and only if, the equality Uj+k-1 — ui — ui+2k-3 — Ui+k2 is satisfied. Proof. In expression (1) only the coefficients of the terms (dj_i — d;) and (dj +fc_2 — di+ik-3) depend 011 A, the rest of the sum can be considered as a constant translation vector which we denote by p. (i) If S = u,-_|_jfc_i — Ui = Uj +2i;-3 — Ui+k2 then 1/(6 — A ) can be factored out, thus we obtain a straight line of the form s i+ k-2 K A) = P + J^— ((u i +k_1 - u) Nf~ 1 (u) (dj _ 1 - di) + (u - Ui+k2) Ni+k2 ( u ) (d i +k2 - d i +k-3)) • (ii) If Ui+k1 — Ui ^ Ui+2k —3 — Ui+k2 then the rational curve (1) (in A) has two points at infinity, one at A = iti+k-i — Ui, and another at A = Ui+2k-3~Ui+k-2) therefore the curve can not be a straight line.
