Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2004. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 31)

TORNAI , R. , Shape modification of cubic B-spline curves by means of knot pairs

64 R. Tornai where N?2(v) Nf(v) = [v

j

+1 (®i+l — ®j-l)(

vi+l - VJ (v - Vj {Vj+2 - Vj){Vj+1 - Vj) 3.1. Examined areas so far In [1], [4] and [7] three pairs of knots are allowed to change. These are (vj-i, Vj), (vj,vj+1) and (vj+i, ^j+i)- The corresponding permissible regions of p will be denoted by J?i, i?

2 and i?3 respectively. (In this case the aim is to minimize the number of altering arcs of s(w), so only the change of consecutive knots are allowed.) The boundary of sub regions i? 1, and i?3 are formed by paths that belong to different extreme positions of the point h(i). i?i is bounded by three paths. The first path is determined by letting 2 = Vj1 and varying Vj

; the second by letting vj — Vj and varying and the third path is determined by letting t'j_i = Vj and varying them simultaneously. Qi is bounded by four paths. The first path is determined by letting Vj +

1 = Vj-\-2 and varying Vj; the second by letting Vj+i = Vj and varying Vj the third by letting Vj — Vj1 and varying Vj +

1 and the fourth path is determined by letting Vj — Vj and varying Uj+i. i?3 is bounded by three paths. The first path is determined by letting Vj+

2 = Vj+ 3 and varying Vj+

1; the second by letting Vj+

1 — Vj and varying Vj+

2 and the third path is determined by letting Vj

+

1 = Vj+

2 and varying them simultaneously. These three overlapping regions are shown in Fig. 1. a), d), e). Thus, if the point p is in the union of these three regions above, then the solution to the shape modification problem s(ii) —> p is guaranteed. In such a case, the number of solutions can be 1, 2 or 3 depending on the position of p with respect to the regions and i?

3. In order to obtain the solutions, we have to solve the system of equations K + l - Vj-l){Vj + 1 - Vj) ^ y

= (fi-Pj)

2 Oj+2 - Vj)(Vj + l - Vj) either for the pair of unknowns (vj-i,vj) or for (uj, t>j+i) or for (vj+i, ^+2)' Only those solutions of Eq. (2) provide solutions to the shape modification problem which fulfills the monotonicity condition v3-1 < v3 < 0 < + 1 <