Line by two planes intersection

Question

I've been wondering for a long time and I can't find, where is my fault.

I write simple program to compute line of two planes intersection.

function getIntersectOf2Planes ( $P1 , $P2 )
{
      /*  Line equation in parametric form:
                x = x0 + t*a
                y = y0 + t*b
                z = z0 + t*c
            */
        $x0 = ( $P1->B * $P2->D - $P2->B * $P1->D ) / ( $P1->A * $P2->B - $P2->A * $P1->B ) ;
        $a = ( $P1->B * $P2->C - $P2->B * $P1->C );
        $y0 = ( $P2->A * $P1->D - $P1->A * $P2->D ) / ( $P1->A * $P2->B - $P2->A * $P1->B ) ;
        $b = ( $P2->A * $P1->C - $P1->A * $P2->C );
        $z0 = 0;
        $c = 1;
}

I found this formula on site: http://www.ambrsoft.com/TrigoCalc/Plan3D/Plane3D_.htm

But when i run my program for planes: Plane1: -25x -10y +4z +125 = 0 Plane2: 2x +21y -8z -12 =0

I have a result:

x0 = 4.960396039604
y0 = 0.099009900990099
z0 = 0
a  = -4
b  = -192
c  = 1

What means:

x = 4.96 -4t (rounded)
y = 0.1 -192t
z = t

But the online calculators, give me:

x = 4.96 − 4t
y = 0.1 − 192t
z = − 505t

As you can see, the 'c' parametr is a problem.

How to fix it?

//EDIT: Looking on:

$a = ( $P1->B * $P2->C - $P2->B * $P1->C );
$b = ( $P2->A * $P1->C - $P1->A * $P2->C );

I tried:

$c = (  $P1->A * $P2->B - $P2->A * $P1->B );

and it's working. But why? Why on this site above is:

z = t

Answer 1

Yes indeed the formula given is uncorrect, we need

$$z=t(A_1B_2-B_1A_2)$$

indeed the direction vector for the parametric line intersection is obtained by the cross product of the two normal vectors of the planes.