Incidence (geometry) - Collinearity

Collinearity

The dual of concurrency is collinearity. Three points P₁, P₂, and P₃ in the projective plane are collinear if they all lie on the same line. This is true if and only if

but if the points are expressed in homogeneous coordinates then these three different equations can be collapsed into one equation:

which is more symmetrical and whose computation is straightforward.

If P₁ : (x₁ : y₁ : z₁), P₂ : (x₂ : y₂ : z₂), and P₃ : (x₃ : y₃ : z₃), then P₁, P₂, and P₃ are collinear if and only if

i.e. if and only if the determinant of the homogeneous coordinates of the points is equal to zero.