NOTE ✦ Mon Jun 08 2026 00:00:00 GMT+0000 (Coordinated Universal Time)

Notes on projective geometry

A short refresher. Headings below become the TOC automatically.

Vanishing points

Inline math like v∈R2v \in \mathbb{R}^2 works mid-sentence. Parallel lines in the world meet at a single point under perspective projection.

MProj-IoU=1N∑i=1N∣P^i∩Pi∣∣P^i∪Pi∣\mathrm{MProj\text{-}IoU} = \frac{1}{N}\sum_{i=1}^{N} \frac{|\hat P_i \cap P_i|}{|\hat P_i \cup P_i|}

Two world-parallel lines share a vanishing point; their image directions are no longer parallel.

Rotations

A 2-D rotation is orthogonal, so its transpose is its inverse:

rotationR(θ)=[cos⁡θ−sin⁡θsin⁡θcos⁡θ],R⊤R=I\mathbf{R}(\theta)=\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix},\quad \mathbf{R}^\top\mathbf{R}=\mathbf{I}

A note on code

import numpy as np
def rotate(t):
    c, s = np.cos(t), np.sin(t)
    return np.array([[c, -s], [s, c]])

Recap

  • Inner product → alignment; its root → length.
  • Vanishing points come from projective geometry, not perspective tricks.