Image Secret Sharing - Visual Cryptography and Secret Image Sharing - page 386

0; t; 2t; ; 1 are on a curve. The parameter t = 0; t; 2t; ; 1 is adap-
tively given and it may take any real number value in the interval [0,1] for a
piece of curve and it also can be extended to the innite interval [1; +1]
for the whole curve. The curve turns out to be a conic section, which can be
expressed as a rational Bernstein{Bezier curve P[t] as follows:

P[t] = L 0 [t] L 1 [t] (14.17)

It is clear that P[t] is a quadratic rational Bezier curve [27][28] whose
control points are:

P 0 = L 0;0 L 1;0 ;P 1 = L 0;0 L 1;1 + L 0;1 L 1;0
2

;P 2 = L 0;1 L 1;1 (14.18)

The graph of a quadratic curve generated by moving lines is shown in
Figure 14.5(a).

(a) The graph of a quadratic curve (b) The graph of a quartic curve

FIGURE 14.5
Intersection of two pencils of lines in (a) and (b).

In general, the curve comprising of L 0;0 , L 0;1 , L 0;2 , and L 1;0 , L 1;1 is:

L 0 [t] = L 0;0 (1t) + 2L 0;1 (1t)t + L 0;2 t 2 ;L 1 [t] = L 1;0 (1t) + L 1;1 t (14.19)

then P[t] = L 0 [t] L 1 [t], t 2 [0; 1] is a cubic rational Bezier curve [16].
Without loss of generality, the moving lines consisting of L 0;0 , L 0;1 , ,
L 0;p , and L 1;0 , L 1;1 , , L 1;q are:

p X q X

L i;0 B i (t);L 1 [t] = L i;1 B i (t) L 0 [t] = (14.20)

i=0 i=0