ex. in 2D (1D projection)
\(y' = x'\,\frac{y}{x} = f\;\frac{y}{x} \;\;\;\) (\(f\): focal)
Linear model using 3D vectors in projective space. \[\small p' = \left( \begin{array}{c}f \\ f \frac{y}{x} \\ 1
\end{array} \right) \underbrace{=}_{normalization} \left( \begin{array}{c}f x \\ f y \\ x \end{array} \right) = \left( \begin{array}{ccc} f & 0 & 0 \\ 0
& f & 0 \\ 1 & 0 & 0 \end{array} \right) \left( \begin{array}{c} x \\ y \\ 1 \end{array} \right) \] considering that the last coordinate must always be
normalized to 1 (for points).
- Projective space
-
- - Real points lie on \(z=1\)
- - Vectors lie on \(z=0\)
- Real coordinates of points are obtained after normalization (division by z).