2D and 3D Transformations

Euclidean, similarity, affine, and projective transformations in 2D and 3D, represented as homogeneous matrices.

Geometric transformations map points from one coordinate frame to another. Representing them as homogeneous matrices allows any chain of transformations to be composed by matrix multiplication, and makes the hierarchy of transformation groups explicit.

Hierarchy of 2D transformations

Each class of 2D transformation preserves a strictly smaller set of geometric properties than the one below it.

Class	DOF	Preserved	Matrix form
Translation	2	Distances, angles	$\begin{pmatrix}I & \mathbf{t}\\ \mathbf{0}^\top & 1\end{pmatrix}$
Euclidean (rigid)	3	Distances, angles	$\begin{pmatrix}R & \mathbf{t}\\ \mathbf{0}^\top & 1\end{pmatrix}$
Similarity	4	Angles, ratios of lengths	$\begin{pmatrix}sR & \mathbf{t}\\ \mathbf{0}^\top & 1\end{pmatrix}$
Affine	6	Parallelism, area ratios	$\begin{pmatrix}A & \mathbf{t}\\ \mathbf{0}^\top & 1\end{pmatrix}$
Projective (homography)	8	Cross-ratio, collinearity	$\begin{pmatrix}\tilde{A} & \tilde{\mathbf{t}}\\ \mathbf{v}^\top & v\end{pmatrix}$

Translation

$$\mathbf{m}' = T\,\mathbf{m}, \qquad T = \begin{pmatrix}1 & 0 & t_x\\ 0 & 1 & t_y\\ 0 & 0 & 1\end{pmatrix}$$

Rotation by angle $\theta$

$$R(\theta) = \begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}$$

The full rigid-body matrix in homogeneous form:

$$H_{\text{Eucl}} = \begin{pmatrix}\cos\theta & -\sin\theta & t_x\\ \sin\theta & \cos\theta & t_y\\ 0 & 0 & 1\end{pmatrix}$$

Affine transformation

$$H_{\text{aff}} = \begin{pmatrix}a_{11} & a_{12} & t_x\\ a_{21} & a_{22} & t_y\\ 0 & 0 & 1\end{pmatrix}$$

Parallel lines remain parallel under affine maps; angles and lengths in general do not.

Projective (homography)

$$H = \begin{pmatrix}h_{11} & h_{12} & h_{13}\\ h_{21} & h_{22} & h_{23}\\ h_{31} & h_{32} & h_{33}\end{pmatrix}$$

defined up to scale (8 degrees of freedom). Straight lines map to straight lines, but parallelism is not preserved.

3D transformations

In 3D the same hierarchy applies. A rigid-body transformation uses a $3\times3$ rotation matrix $R$ and a translation vector $\mathbf{t}\in\mathbb{R}^3$:

$$\mathbf{M}' = \begin{pmatrix}R & \mathbf{t}\\ \mathbf{0}^\top & 1\end{pmatrix}\mathbf{M}$$

Rotation matrices and the Rodrigues formula

Any 3D rotation can be parameterised by an axis $\hat{\mathbf{n}}$ (unit vector) and an angle $\theta$. The Rodrigues rotation formula gives:

$$R = I\cos\theta + (1-\cos\theta)\hat{\mathbf{n}}\hat{\mathbf{n}}^\top + \sin\theta\,[\hat{\mathbf{n}}]_\times$$

where $[\hat{\mathbf{n}}]_\times$ is the skew-symmetric cross-product matrix:

$$[\mathbf{n}]_\times = \begin{pmatrix}0 & -n_z & n_y\\ n_z & 0 & -n_x\\ -n_y & n_x & 0\end{pmatrix}$$

In MATLAB, the compact Rodrigues vector $\boldsymbol{\omega} = \theta\hat{\mathbf{n}}$ encodes both axis and angle in 3 numbers, and the conversion is available via rodrig2R / R2rodrig.

3D-to-2D perspective projection

The full camera projection matrix $P$ maps a 3D world point $\mathbf{M}$ to a 2D image point $\mathbf{m}$:

$$\mathbf{m} \sim P\,\mathbf{M}, \qquad P = K\,[R\;|\;\mathbf{t}]$$

where $K$ is the intrinsic matrix (see Camera Calibration):

$$K = \begin{pmatrix}f_x & s & c_x\\ 0 & f_y & c_y\\ 0 & 0 & 1\end{pmatrix}$$

MATLAB examples

2D transformations (Trans2D)

% Interactive 2D transformation demo using Balu Toolbox
% 1. Go to the folder containing painting.png
%    cd ../images
% 2. Launch the GUI
Bmv_guihomography
% 3. Press Load and type 'painting.png'
% 4. Use the GUI buttons to apply:
%    - rotation, translation, similarity, affine, projective

Python examples

3D transformations (Colab)

Notebook covering rotation matrices, the Rodrigues formula, and 3D-to-2D perspective projection in Python.

Video lecture

Lecture: 2D transformations and homographies (2021)

Recorded class covering the full hierarchy of 2D transformations and their homogeneous matrix representations.

Key properties at a glance

Composing transformations

Because every transformation is a matrix, composing two transformations is just matrix multiplication: $H_{\text{total}} = H_2 H_1$. The order matters — matrix multiplication is not commutative in general.

Inverting a rigid-body transformation

For a rigid-body matrix $(R,\mathbf{t})$ the inverse is $(R^\top, -R^\top\mathbf{t})$, since rotation matrices are orthogonal ($R^{-1} = R^\top$). This is much cheaper than a general matrix inversion.

Degrees of freedom count

Translation: 2 (tx, ty). Euclidean: 3 (+angle). Similarity: 4 (+scale). Affine: 6 (full 2×2 matrix + translation). Projective: 8 (3×3 matrix up to scale). In 3D, rigid-body has 6 DOF (3 rotation + 3 translation).

What is preserved under each transformation?

Euclidean: distances and angles. Similarity: angles and length ratios. Affine: parallelism, ratios of collinear lengths, area ratios. Projective: cross-ratio of four collinear points, and collinearity.