I have two points $P_1=(x_1,y_1,z_1)$, and $P_2=(x_2,y_2,z_2)$, also I have my plane values $A,B,C $ and $D$ too.
I know that $P_1$ lies on a side of the plane, and $P_2$ lies on other side of the plane, so there is an interesection with these two points with the plane.
Here is the algortihm to get the intersection point:
$u = Ax_1 + By_1 + Cz_1 + D$
$v = Ax_2 + By_2 + Cz_2 + D$
$ratio = u / (u - v )$
So, the intersection point is calculated using the parametric form of the line:
$xi = x_1 + ratio( x_2 - x_1 )$
$yi = y_1 + ratio( y_2 - y_1 )$
$zi = z_1 + ratio( z_2 - z_1 )$
I don't understand this operation: $ratio = u / (u - v )$, what thing the values $u$ and $v$ are telling me?, why not calculating the ratio like this $ratio = v / (v - u )$ ??? Why they can be used as ratio for the parametic form of the line to get the intersection point?