Basic Primitive Tests - Real-Time Collision Detection - page 209

int IntersectPlanes(Plane p1, Plane p2, Point &p, Vector &d)
{

// Compute direction of intersection line
d = Cross(p1.n, p2.n);

// If d is zero, the planes are parallel (and separated)
// or coincident, so they're not considered intersecting
if (Dot(d, d) < EPSILON) return 0;

float d11 = Dot(p1.n, p1.n);
float d12 = Dot(p1.n, p2.n);
float d22 = Dot(p2.n, p2.n);

float denom = d11*d22 - d12*d12;
float k1 = (p1.d*d22 - p2.d*d12) / denom;
float k2 = (p2.d*d11 - p1.d*d12) / denom;
p = k1*p1.n + k2*p2.n;
return 1;

}

This code can be optimized by realizing that the denominator expression is really
just the result of applying Lagrange's identity to ( n 1 × n 2 ) · ( n 1 × n 2 ). The denominator

is therefore equivalent to

denom = d · d .

Using the vector identity

u × ( v × w ) = ( u · w ) v − ( v · w ) u ,

the expression for P = k 1 n 1 + k 2 n 2 can also be further simplified as follows.

P = k 1 n 1 + k 2 n 2 ⇔ (original expression)

P =[ ( d 1 ( n 2 · n 2 ) − d 2 ( n 1 · n 2 )) n 1 + ( d 2 ( n 1 · n 1 ) − d 1 ( n 1 · n 2 )) n 2 ⇔

(substituting k 1 and k 2 )

] / denom

P denom = ( d 1 ( n 2 · n 2 ) − d 2 ( n 1 · n 2 )) n 1 + ( d 2 ( n 1 · n 1 ) ⇔

(multiplying both sides by denom)

− d 1 ( n 1 · n 2 )) n 2

P denom = ( d 1 ( n 2 · n 2 ) n 1 − d 2 ( n 1 · n 2 ) n 1 ) + ( d 2 ( n 1 · n 1 ) n 2 ⇔

(distributing scalar multiply)

− d 1 ( n 1 · n 2 ) n 2 )