Affine Geometry

🧭 Introduction

Affine Geometry studies geometric transformations that preserve points, straight lines, and planes. While lengths and angles may change, parallelism and ratios along lines remain invariant. This makes affine geometry fundamental in computer vision and rendering systems.

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2D Affine Geometry

Translation

A point $P=(p_1,p_2{)}^T$ is translated by a vector $T=(t_1,t_2{)}^T$:

$$P^{\prime }=P+T=\left[\begin{array}{c}{p}_1+t_1\\ p_2+t_2\end{array}\right]$$

Used when coordinate systems shift position but retain orientation.

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Rotation

If both systems have the same origin and differ only in orientation:

$$P^{\prime }=R(\phi )\cdot P$$

With:

$$R(\phi )=\left[\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{array}\right]$$

👉 Properties of an orthonormal matrix $R$: $R^{-1}=R^T$

• Length and angles are preserved

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Scaling

Uniform or non-uniform stretching:

$$P^{\prime }=S\cdot P,S=\left[\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right]$$

Can distort angles and lengths.

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Shearing

Deforms the shape by slanting:

$$P^{\prime }=S\cdot P,S=\left[\begin{array}{cc}1& s_1\\ s_2& 1\end{array}\right]$$

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Reflections

Several common 2D reflection matrices:

• About $y=x$: $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$
• About $y=-x$: $\left[\begin{array}{cc}0& -1\\ -1& 0\end{array}\right]$
• About x-axis: $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$

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General Form of Affine Transformations

$$P^{\prime }=A\cdot P+T$$

Where:

• $A\in {\mathbb{R}}^{2\times 2}$ : Linear part
• $T\in {\mathbb{R}}^2$ : Translation vector

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Properties

1. Affine invariance: Ratios and linear combinations are preserved
2. Parallelism preserved
3. Lengths and angles may change
4. 3 non-collinear point pairs are needed to define the transformation

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Similarity Transformations

$$P^{\prime }=s\cdot R\cdot P+T$$

Where:

• $s$: Scaling factor
• $R$: Rotation matrix
• $T$: Translation vector

Preserves: ratios, angles, and parallel lines

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Rigid Body Motion

$$P^{\prime }=A\cdot P+T,\text{with }A\text{ orthogonal}$$

Preserves:

• Ratios
• Angles
• Lengths
• Parallelism

Reflection allowed if $\det (A)=-1$

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Homogeneous Coordinates

Lift 2D points into 3D for unified matrix transformations:

$$(x,y{)}^T\to (x,y,1{)}^T$$

Affine transformations as 3×3 matrices:

Translation:

$$\left[\begin{array}{ccc}1& 0& t_1\\ 0& 1& t_2\\ 0& 0& 1\end{array}\right]$$

Rotation:

$$\left[\begin{array}{ccc}\cos \phi & -\sin \phi & 0\\ \sin \phi & \cos \phi & 0\\ 0& 0& 1\end{array}\right]$$

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3D Affine Geometry

Translation, Scaling

Translation Matrix:

$$T(tx,ty,tz)=\left[\begin{array}{cccc}1& 0& 0& tx\\ 0& 1& 0& ty\\ 0& 0& 1& tz\\ 0& 0& 0& 1\end{array}\right]$$

Scaling Matrix:

$$S({\lambda }_1,{\lambda }_2,{\lambda }_3)=\left[\begin{array}{cccc}{\lambda }_1& 0& 0& 0\\ 0& {\lambda }_2& 0& 0\\ 0& 0& {\lambda }_3& 0\\ 0& 0& 0& 1\end{array}\right]$$

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Rotation Around Coordinate Axes

$R_z(\phi )$ – Rotation around Z-axis:

$$\left[\begin{array}{cccc}\cos \phi & -\sin \phi & 0& 0\\ \sin \phi & \cos \phi & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(Analogous for $R_x(\phi )and{R}_y(\phi ))$

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Rotation Around Arbitrary Axis

1. Rotate the axis to align with z-axis
2. Apply $R_z(\psi )$
3. Invert the alignment transformation

Final matrix:

$$M_b(\psi )=T(a)\cdot R_z(\theta )\cdot R_y(\varphi )\cdot R_z(\psi )\cdot R_y(-\varphi )\cdot R_z(-\theta )\cdot T(-a)$$

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General Form of 3D Affine Transformation

$$P^{\prime }=A\cdot P+T\text{or in homogeneous coordinates:}{P}^{\prime }=M\cdot P$$

Where $A\in {\mathbb{R}}^{3\times 3},T\in {\mathbb{R}}^3,M\in {\mathbb{R}}^{4\times 4}$

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Sources

• Hartley & Zisserman, Multiple View Geometry in Computer Vision, 2nd ed.
• Wikipedia: Affine Transformation, Homogeneous Coordinates
• Szeliski, R. (2010). Computer Vision: Algorithms and Applications