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Perspective and projective space

Modeling perspective projection requires division.

ex. in 2D (1D projection)

2d 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).