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
74 Imre Juhász Obviously, paths of its points are elements of the pencil of lines the base point of which is d z, provided ±2 — Ui = 3 — Wj+i» cf. Fig. 1. (This special case can be found in [3] as well, along with other knot-based shape modifications.) This means that the symmetric translation of knots U{ and Ui+2k-z pulls from / pushes toward the control point d 7; points of the arc s,-+i (u ) along straight lines. Therefore this shape modification effect is similar to the one obtained by the alteration of the weigh W{ (the weight of the control point d;). However, these two shape modification methods are not substitutes of each other. The main difference is that only shape of arcs s; (u) , s J +i (u) ,..., Si+k-i (u) are modified when W{ is changed, whereas with the symmetric translation of knots u; and ti,-+2fc-3 arcs (u) , Sj Ar +2 (w) ) • • •, Sj+3fc-5 («) are modified, i.e. in the latter case a much larger portion of the curve is altered. The difference between these two shape modification methods are illustrated in Fig. 2. 2.2. Case k = 4 In this case the arc Si+ 2 (u, A) is of interest which has the form Si+2 («, A) = ( Ui+ 4~ U N? + 1 («) + TV j 3 (u)) d, \ ui+ 4 Ui + l ( u, s'+l \ Ui1-4 ' Ui + l Ui+3 u Ui+3 Ux ; - A u — u i+2 Ui1-5 A - Ui. + ( 7 —('") + Ni+2 («) ) + —— TV 3 (u) (di_i - d,-) + R^ —N? + 2 (U) (D I+ 2 - d i+ 1) . *" - i+2 The coefficients of d; and d, + 1 ai'e non-negative and sum to 1, i.e. the constant part of the sum is a convex linear combination of the control points d; and d 2+ 1. Therefore paths of the arc are straight line segments the extension of which intersect the side d ?, d i + i at its inner points moreover, they are parallel to the plane determined by the directions dj_i — d, és &i+ 2 — d; +i, provided Ui +3 — Ui = «2+5 — Ui + 2 holds, cf. Fig. 3. If the directions d,_i — dj and d; +2 _ dj+i are parallel then the paths form a pencil of parallel lines.
