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
Shape modification of' cubic B-spline curves by means of knot pairs 63 has an envelope which is a B-spline curve of order (k — 1) and can be written in the form i- 1 h(v) = Y, diNf'^v), v G [f>i-i,t>i] l=i-k + 1 where the knot values _iuj , ifj<i : l \ My + i, otherwise Vj i.e., from the knot values {itj } we have to leave out the ith one, where the multiplicity of Ui is one. Their points of contact are h («,•) = s(// 2-, «,•). For k = 4 by the modification of the knot value u,+i, we obtain a oneparameter family of cubic B-spline curves of the form n s («, u j+ l) = diN* (u, Uj+i) , u G [u 3, u n +i] , u j +i G [uj, u j+ 2) with knots wo, u\, . .. u n + 4, and the envelope is the parabolic arc j 1=3-2 with knots Vj-2 = «j-2,^-1 = «j-uVj = Uj,v j+ l = u j+ 2, v j+ 2 = «j+ 3,Wj+3 = "j+43. Move a point of the curve to a specified location A generally accepted shape modification method is, when the user picks a point of the curve, then species a new location where the picked point has to be moved. Furthermore, let's assume that for the parameter of the picked point s(w), ű G [«,•,«,•+2) holds. The new location will be denoted by p, and its coordinates in the coordinate system {d/_i; d/_2 — 1; d ; — d ;_i} by x and y. It is known (c.f. [1], [4], [7]) that the s (ü) —> p shape modification can be performed by the alteration of three consecutive knots of the curve s(u). For the determination of the permissible positions of p the following has to be taken into account: in the fíézier representat ion of the envelope the value t which corresponds to v — ü varies with the variation of the knots Vj and fj + i, since t = (v — Vj)/('Vj + i — Vj). Therefore, the B-spline representation of the envelope can be used. Utilizing that Nf_ 2{v) + Nf_i{v) + Nf(v) = l,Vv G [vj,v j+ 1) Eq. (1) can be written in the form h» = dj _ 1 + Nj_ 2(v)(dj2 - dj_!) + Nj(v)(dj - d,_,)
