---
date: '2025-09-15'
description: A linear map plus translation expressed with homogeneous coordinates
id: affine transformation
modified: 2026-09-01 09:15:00 GMT-04:00
tags:
- seed
title: affine transformation
created: '2025-09-15'
published: '2025-09-15'
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slug: thoughts/affine-transformation
permalink: https://aarnphm.xyz/thoughts/affine-transformation.md
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---
> \[!abstract\] What is an affine transformation?
>
> An affine transformation is a map of the form
> $\displaystyle f(x) = A\,x + b,$
> where $A\in\mathbb{R}^{m\times n}$ is a linear map and $b\in\mathbb{R}^m$ is a translation vector. An affine map sends a line to a line or a point and preserves convex combinations. When $A$ is injective, it also preserves parallelism and ratios of lengths along a line. Translation makes affine maps more general than linear maps.
An affine map acts on points. Choosing coordinate origins represents those points as vectors; see [[thoughts/Vector space]]. A map from $\mathbb R^n$ to $\mathbb R^m$ has $m(n+1)$ parameters: $mn$ entries in $A$ and $m$ entries in $b$. An affine endomap on $\mathbb R^n$ therefore has $n^2+n$ parameters.
## standard form
- Definition: $f: \mathbb{R}^n\to\mathbb{R}^n$, $f(x)=A x + b$.
- Composition: for $f(x)=A_f x + b_f$ and $g(x)=A_g x + b_g$,
$\displaystyle (f\circ g)(x) = A_f(A_g x + b_g) + b_f = (A_f A_g)\,x + (A_f b_g + b_f).$
- Invertibility: $f$ is invertible iff $A$ is invertible. The inverse is
$\displaystyle f^{-1}(y) = A^{-1}(y - b).$
- Fixed point: a point $x_\star$ with $f(x_\star)=x_\star$ satisfies
$\displaystyle (I - A) x_\star = b, \quad x_\star = (I-A)^{-1} b \;\text{ if }\; I-A \text{ is invertible}.$
## homogeneous coordinates
Affine maps become linear in one higher dimension using homogeneous coordinates. Define $\tilde{x} = \begin{bmatrix}x\\1\end{bmatrix}$ and
$\displaystyle \tilde{A} = \begin{bmatrix} A & b \\ 0 & 1 \end{bmatrix}. $
Then $\tilde{A}\,\tilde{x} = \begin{bmatrix} A x + b \\ 1 \end{bmatrix}$ encodes $f(x)=Ax+b$. Composition reduces to matrix multiplication of the $\tilde{A}$ blocks.
## geometric properties
- Linearity on barycentric combinations: for scalars $\{\lambda_i\}$ with $\sum_i \lambda_i = 1$,
$\displaystyle f\Big(\sum_i \lambda_i x_i\Big) = \sum_i \lambda_i f(x_i).$
- Affine maps preserve barycentric identities such as midpoints and centroids. Images of collinear points remain collinear. An injective affine map also preserves distinct parallel lines and ratios along a line. General lengths and angles can change.
- For a square linear part $A$, $n$-dimensional volume scales by $|\det A|$. An invertible map preserves orientation when $\det A>0$ and reverses it when $\det A<0$.
## common 2d/3d examples
- Translation: $A=I$, $f(x)=x+b$.
- Uniform scaling by $s$: $A=s I$, $f(x)=s x$.
- Anisotropic scaling: $A=\operatorname{diag}(s_1,\dots,s_n)$.
- Rotation (2D):
$\displaystyle A=\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix},\quad f(x)=Ax.$
- Shear (2D, $x$-shear by $k$): $A=\begin{bmatrix}1 & k\\ 0 & 1\end{bmatrix}$.
- Reflection across the origin-centered hyperplane with unit normal $u$: $A=I-2uu^\top$. For the affine hyperplane $u^\top x=c$, the reflection is $f(x)=x-2(u^\top x-c)u$.
## algebraic structure
- The set of invertible affine transformations on $\mathbb{R}^n$ forms the affine group $\operatorname{GA}(n) \cong \operatorname{GL}(n) \ltimes \mathbb{R}^n$ (semidirect product of the general linear group with translations).
- Composition and inversion follow the block rules from the standard form. Determinants and traces refer to the linear part $A$.
## fitting an affine map
Given paired points $\{(x_i,y_i)\}_{i=1}^N$ with $x_i,y_i\in\mathbb{R}^n$, an affine map $y\approx Ax+b$ can be estimated via least squares by augmenting inputs $\hat{x}_i = [x_i^\top,\;1]^\top$ and solving
$\displaystyle \min_{\tilde{A}\in\mathbb{R}^{n\times(n+1)}}\; \sum_i \big\|y_i - \tilde{A} \, \hat{x}_i\big\|_2^2,$
where $\tilde{A}=[A\;\;b]$.
## invariants and non-invariants
- Every affine map preserves convexity, barycentric coordinates, and centroids. An injective affine map also preserves distinct parallel lines and ratios of lengths along a common line.
- General affine maps can change angles, absolute lengths, circles, and orthogonality. An orthogonal $A$ preserves lengths and angles. A similarity $A=sQ$, with $s>0$ and $Q$ orthogonal, preserves angles and maps circles to circles while scaling all lengths by $s$.
## notes
- When $A$ is orthogonal with $\det A=1$, $f$ is an orientation-preserving rigid motion that preserves distances.
- When $A=sR$ with $R$ orthogonal and $s>0$, $f$ is a similarity. It preserves angles and scales lengths by $s$; $R$ may include a reflection.
Affine shear + translation on a unit square ($A=\begin{bmatrix}1 & 0.6\\0 & 1\end{bmatrix}$, $b=\begin{bmatrix}1.2\\0.4\end{bmatrix}$):
source code
Rotation + translation on a triangle ($A=R_\theta$, $\theta=25^\circ$, $b=(1.0,0.6)$):
source code
