Homogeneous Coordinates

Definition

A 2D point $(x,y)$ is represented by a 3D point $(x^{\prime },y^{\prime },z^{\prime })$ with an extra “fictitious” coordinate. The 2D point is recovered by dividing by the last coordinate:

$$x=\frac{x^{\prime }}{z^{\prime }}y=\frac{y^{\prime }}{z^{\prime }}$$

Defined up to scale

$$(x,y)\widehat{=}(x,y,1)=(kx,ky,k)\text{for any }k\ne 0$$

Intuition

Perspective projection divides by $Z$, so it can’t be a matrix in normal coordinates. In homogeneous coordinates the matrix product stays linear, and the division moves to the very end (dividing by $z^{\prime }$). Translations also become matrix products, so a whole chain of transformations collapses into one matrix.

Geometrically, a homogeneous point is a ray through the origin. Dividing by $z^{\prime }$ means intersecting this ray with the plane $z^{\prime }=1$.

Appears in

• Lecture 2.1, Homogeneous Coordinates: perspective matrix equation
• Lecture 2.1, Extrinsic parameters: translation as a 4x4 matrix

Related concepts

• Projection Matrix
• Pinhole Camera Model