[1] R. Szeliski, Computer Vision: Algorithms and Applications, 2010.
[2] R. Szeliski, A. Hai-zhou, Computer Vision: Algorithms and Applications, Tsinghua University Press, 2012.

1. Geometric Primitives and Transformations

1.1 Geometric Primitives

2D points. A 2D point can be represented in Cartesian form \(\mathbf{x} = (x, y) \in \mathbb{R}^2\) or in homogeneous coordinates \(\tilde{\mathbf{x}} = (\tilde{x}, \tilde{y}, \tilde{w}) \in \mathbb{P}^2\) with \(\mathbb{P}^2 = \mathbb{R}^3 \setminus \{(0,0,0)\}\). The homogeneous vector relates to its Cartesian counterpart through \(\tilde{\mathbf{x}} = (\tilde{x}, \tilde{y}, \tilde{w}) = \tilde{w}(x, y, 1) = \tilde{w}\,\bar{\mathbf{x}},\) where \(\bar{\mathbf{x}} = (x, y, 1)\) is the augmented vector.

2D lines. A line expressed in homogeneous coordinates \(\tilde{\mathbf{l}} = (a, b, c)\) satisfies \(\bar{\mathbf{x}} \cdot \tilde{\mathbf{l}} = ax + by + c = 0\). The intersection of two lines is \(\tilde{\mathbf{x}} = \tilde{\mathbf{l}}_1 \times \tilde{\mathbf{l}}_2\), whereas the line through two points is \(\tilde{\mathbf{l}} = \tilde{\mathbf{x}}_1 \times \tilde{\mathbf{x}}_2\). Least-squares fitting recovers a line from noisy points or the best intersection of almost-concurrent lines.

2D conics. Conics are described by \(\tilde{\mathbf{x}}^{T} Q \tilde{\mathbf{x}} = 0\) and play a key role in multi-view geometry and camera calibration.

3D points. Analogously, a 3D point can be written as \(\mathbf{x} = (x, y, z) \in \mathbb{R}^3\) or \(\tilde{\mathbf{x}} = (\tilde{x}, \tilde{y}, \tilde{z}, \tilde{w}) \in \mathbb{P}^3\) with augmented form \(\bar{\mathbf{x}} = (x, y, z, 1)\) so that \(\tilde{\mathbf{x}} = \tilde{w}\,\bar{\mathbf{x}}\).

3D planes. A plane with homogeneous parameters \(\tilde{\mathbf{m}} = (a, b, c, d)\) satisfies \(\bar{\mathbf{x}} \cdot \tilde{\mathbf{m}} = ax + by + cz + d = 0\).

3D lines. A line defined by two points \((\mathbf{p}, \mathbf{q})\) can be parameterized as \(\mathbf{r} = (1-\lambda)\mathbf{p} + \lambda \mathbf{q}\) with \(0 \le \lambda \le 1\). Homogeneously, \(\tilde{\mathbf{r}} = \mu \tilde{\mathbf{p}} + \lambda \tilde{\mathbf{q}}\), which has excess degrees of freedom. Plücker coordinates remedy this with \(\mathbf{L} = \tilde{\mathbf{p}}\,\tilde{\mathbf{q}}^{T} - \tilde{\mathbf{q}}\,\tilde{\mathbf{p}}^{T},\) subject to \(\det(\mathbf{L}) = 0\), leaving four degrees of freedom.

3D quadrics. Quadrics mirror conics via \(\tilde{\mathbf{x}}^{T} Q \tilde{\mathbf{x}} = 0\).

1.2 Two-Dimensional Transformations

Planar transformations include translation, Euclidean transforms (rotation plus translation), similarity transforms (uniform scale + Euclidean), affine transforms, and projective transforms.

Translation. \(x' = x + t, \qquad x' = [\,\mathbf{I} \ \ t\,]\bar{x},\) or in homogeneous form, \(\bar{x}' = \begin{bmatrix} \mathbf{I} & t \\ 0^{T} & 1 \end{bmatrix} \bar{x},\) which enables chaining via \(3\times3\) matrices.

Euclidean transformation (rotation + translation). \(x' = \mathbf{R}x + t = [\,\mathbf{R} \ \ t\,]\bar{x},\) where \(\mathbf{R} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\) is an orthogonal rotation matrix satisfying \(\mathbf{R}\mathbf{R}^{T} = \mathbf{I}\) and \(|\mathbf{R}|=1\).

Similarity transformation (scale + rotation + translation). \(x' = s\mathbf{R}x + t = [\,s\mathbf{R} \ \ t\,]\bar{x} = \begin{bmatrix} a & -b & t_x \\ b & a & t_y \end{bmatrix}\bar{x},\) with \(a^2 + b^2 = s^2\). Similarities preserve angles between lines.

Affine transformation. \(x' = \mathbf{A}\bar{\mathbf{x}}, \qquad x' = \begin{bmatrix} a_{00} & a_{01} & a_{02} \\ a_{10} & a_{11} & a_{12} \end{bmatrix}\bar{x},\) where \(\mathbf{A}\) is \(2\times3\). Affine maps preserve parallelism.

Projective transforms extend this framework by allowing arbitrary \(3 \times 3\) homogeneous matrices and are handled similarly, though they are not detailed further here.