The intersection of two planes

Written by Paul Bourke
February 2000

The intersection of two planes (if they are not parallel) is a line.

Define the two planes with normals N as

N₁ . p = d₁

N₂ . p = d₂

The equation of the line can be written as

p = c₁ N₁ + c₂ N₂ + u N₁ * N₂

Where "*" is the cross product, "." is the dot product, and u is the parameter of the line.

Taking the dot product of the above with each normal gives two equations with unknowns c₁ and c₂.

N₁ . p = d₁ = c₁ N₁ . N₁ + c₂ N₁ . N₂

N₂ . p = d₂ = c₁ N₁ . N₂ + c₂ N₂ . N₂

Solving for c₁ and c₂

c₁ = ( d₁ N₂ . N₂ - d₂ N₁ . N₂ ) / determinant

c₂ = ( d₂ N₁ . N₁ - d₁ N₁ . N₂) / determinant

determinant = ( N₁ . N₁ ) ( N₂ . N₂ ) - ( N₁ . N₂ )²

Note that a test should first be performed to check that the planes aren't parallel or coincident (also parallel), this is most easily achieved by checking that the cross product of the two normals isn't zero. The planes are parallel if

N₁ * N₂ = 0