Standard Euclidean geometry is built on the assumption that parallel lines stay equidistant forever. However, the human eye and camera sensors don't see the world this way. Parallel railroad tracks appear to converge toward a single point on the horizon.
The Projective Plane ($\mathbb{P}^2$) is an extension of the Euclidean plane that includes "points at infinity." In computer vision, we map points from the 3D world onto 2D sensors using this projective framework.
To perform linear operations in projective space, we represent a 2D point $(x, y)$ as a 3D vector $\mathbf{x} = [x, y, 1]^T$. This 3rd component, $w$, acts as a scale factor. If $w=1$, the point is "on the plane." As $w$ approaches 0, the point moves toward infinity.
| Sym | Definition |
|---|---|
| $\mathbb{P}^n$ | Projective space of dimension $n$ derived from a vector space of dimension $n+1$. |
| $\mathbf{x}$ | A point represented as a $3 \times 1$ column vector in homogeneous coordinates. |
| $\mathbf{l}$ | A vector representing a line in the projective plane such that $\mathbf{l}^T \mathbf{x} = 0$. |
| $w$ | An arbitrary, non-zero homogeneous scale factor. Required to map back to Euclidean coordinates. |