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 65 as well. Such a solution always exists, when p is in t lie region that corresponds to the pair of unknowns, for which the system is solved. 3.2. New areas, that extend the possibilites What is more interesting, we can choose not consecutive knots of the curve. This way we can reach points of three other regions. In this case three pairs of knots are allowed to change also. These are (t'j-i, Vj+1), (fj-i, Vj+2) and (vj

, 1^+2)The corresponding permissible regions of p will be denoted by Q4, and Ű

Q respectively. The boundary of sub regions are formed by paths that belong to different extreme positions of the point h( f>). These new three regions will overlap each other and unfortunately they mean only a little region compared to the union of Í2j, j?2 and 1?

3. Another disadvantage is that the union of i?|

, i?

2 and overlaps mainly the union of Q4, and QQ. In spite of all of t hese facts, these new solutions can be useful. They let greater freedom for the designer to modify the shape of a curve. Here we shall discuss these three regions. The detailed discussion of the permissible positions of the point, the parameter values and the unknowns can be found in [7]. 3.2.1. i?4i the unknowns are Vj

_ 1 and Vj+i The boundaries of the permissible positions of the point p in this case are the paths connecting the following four extreme positions of a point of the quadratic B-spline curve arc b

;

(1?), (t'j-i <E [vj-2,vj] and Vj+\ € [u,Vj+

2]): (1) Vj2 = Vj1 < Vj <U = Vj +

1 < Vj + 2 (2) vj-2 = vj-1 < Vj < u < v

j

+

1 = Vj+2 (3) Vj2 < Vj1 = Vj <U = Vj + i < Vj+2 (4) Vj — 2 < Vj! = Vj <U< Vj + I = Vj + 2

• The paths can be described similarly to the preceding case, but only t hree of them are actual boundaries, the other three paths run inside the region. The boundaries can be seen in Fig. 1. b). 3.2.2. : the unknowns are Vj1 and Vj+2 The boundaries in this case are straight line segments. The paths connect the following extreme positions: (1) Vj — 2 = Vj-1 < Vj < U < V

J+

{ = Vj + 2 < Vj+3 (2) Vj2 = Vj1 < Vj <U< Vj + i < Vj+2 = Vj+3 (3) Vj2 < Uj-1 = Vj <U< Vj +

1 - Vj + 2 < Vj+3