Image Secret Sharing - Visual Cryptography and Secret Image Sharing - page 385

where a, b, and c are not all zero, (X;Y;W) are the homogenous coordinates
of points whose Cartesian coordinates are:

(x;y) = (X=W;Y=W) (14.8)

It is obvious that X;Y;W cannot all be zero. Let P denote the triple (X;Y;W)
and L denote the triple (a;b;c). We refer to the line L that is:

f(X;Y;W)jLP = (a;b;c) (X;Y;W) = aX + bY + cW = 0g (14.9)

Thus, we can see that a point P lies on a line L = (a;b;c) only and if only
P L = 0 , where P L is the dot product.
Now we consider the line L containing two points P 1 = (X 1 ;Y 1 ;W 1 ) and
P 2 = (X 2 ;Y 2 ;W 2 ), and also the point P at which two lines L 1 = (a 1 ;b 1 ;c 1 )
and L 2 = (a 2 ;b 2 ;c 2 ) intersect. Because of the duality principle (of points and
lines), we have these cross products:

L = P 1 P 2 ;P = L 1 L 2 (14.10)

A homogeneous point whose coordinates are functions of a variable t (i.e.,
it is parameterized by a variable t) is denoted as:

P[t] = (X[t];Y [t];W[t]) (14.11)

which actually is the rational curve:

x = X[t]

W[t] ; y = Y [t]

W[t] ;

(14.12)

If the functions are of the following form:

X[t] = X i i [t]; Y [t] = Y i i [t]; W[t] = W i i [t] (14.13)

With f i [t]g being a given set of blending function, then equation (14.11)
denes a curve:

P[t] = X P i i [t] (14.14)

where the homogeneous points are Pi i = (X i ;Y i ;W i ). Likewise,

L[t] = (a[t];b[t];c[t]) (14.15)

denotes the family of lines a[t]x + b[t]y + c[t] = 0.
In order to obtain the intersection of two pencils, we notice the following
four lines L 0;0 , L 0;1 , L 1;0 , and L 1;1 from which two pencils are defined:

L 0 [t] = L 0;0 (1 t) + L 0;1 t;L 1 [t] = L 1;0 (1 t) + L 1;1 t (14.16)

The points, at which they intersect for parameter values t =