Geometric Transformations

Projective

A projective transformation (also known as a projective mapping) is a type of transformation in geometry that maps points from one projective plane to another.

\[x' = H.x\] \[\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} h_{11} & h_{12} & h_{13} \\ h_{21} & h_{22} & h_{23} \\ h_{31} & h_{32} & h_{33} \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}\]

-Preserves: Concurrency, colinearity, order of contact (intersection, tangency, inflection, …), cross ratio

Subsets of projective transformations are: Affine Transformations, Similarity Transformations, Euclidean Transformation, Isometry Transformations

Translation

A translation transformation is a geometric transformation that moves every point of an object or space by the same distance in a specified direction. It is one of the simplest and most fundamental types of transformations, commonly used in computer graphics, robotics, physics, and mathematics.
A 2D translation has 2 degrees of freedom (one for the x-direction and one for the y-direction).
A 3D translation has 3 degrees of freedom (one for the x-direction, one for the y-direction, and one for the z-direction).
To solve for a 2D translation, you need at least 1 point correspondence (PC). Why? One point gives you two equations (one for x and one for y), which is sufficient to solve for the two unknowns ($t_x$ and $t_y$).
To solve for a 3D translation, you need at least 1 point correspondence.

Why? One point gives you three equations (one for x, one for y, and one for z), which is sufficient to solve for the three unknowns ($t_x$, $t_y$, and $t_z$).

\[\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & tx \\ 0 & 1 & ty \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}\]

Affine

-An affine transformation is a geometric transformation that preserves collinearity (points lying on a straight line remain on a straight line) and ratios of distances (the midpoint of a line segment remains the midpoint after transformation).

It includes translation, rotation, scaling, and shearing.
2D Affine Transformation:

Linear Transformation (Rotation, Scaling): 4 DOF

Translation: 2 DOF (translation in x and y directions, $t_x$ and $t_y$).

Total DOF in 2D: 4+2=6 degrees of freedom.

3D Affine Transformation:

Linear Transformation (Rotation, Scaling, Shearing): 9 DOF

Translation: 3 DOF (translation in x, y, and z directions, $t_x$, $t_y$, and $t_z$).

Total DOF in 3D: 9+3=12 degrees of freedom.

Number of PC: 3
Invariants: All - (length, angle, ratio of length)
Preserves: ratio of areas, ratio of parralel lines, parallelism

\[\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & tx \\ a_{21} & a_{22} & ty \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}\]

Similarity

A similarity transformation is a geometric transformation that preserves the shape of an object but may alter its size, orientation, and position. It combines scaling, rotation, and translation into a single transformation.
A subset of affine transformations
2D Similarity Transformation:

Scaling: 1 DOF (uniform scaling factor s).

Rotation: 1 DOF (rotation angle θ).

Translation: 2 DOF (translation in x and y directions, $t_x$ and $t_y$).

Total DOF in 2D: 1+1+2=4 degrees of freedom.

Number of PC: 2
Invariants: All - length
Preserves: ratio of lengths, ratio of areas, angle

\[\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} s\cos\theta & -s\sin\theta & tx \\ \sin\theta & \cos\theta & ty \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}\]

Isometry

An isometry is a geometric transformation that preserves the distances between all pairs of points. In other words, it is a distance-preserving transformation.
A subset of similarity transformations
Isometry = similarity + uniform scaling
2D Isometry:

Rotation: 1 DOF (rotation angle θ).

Translation: 2 DOF (translation in x and y directions, $t_x$ and $t_y$).

Total DOF in 2D: 1+2=3 degrees of freedom.

3D Isometry:

Rotation: 3 DOF (rotation angles around the x, y, and z axes, often represented as Euler angles or a rotation matrix).

Translation: 3 DOF (translation in x, y, and z directions, $t_x$, $t_y$, and $t_z$).

Total DOF in 3D: 3+3=6 degrees of freedom.

Invariants:

\[\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} \epsilon\cos\theta & -\sin\theta & tx \\ \epsilon\sin\theta & \cos\theta & ty \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}\] \[\epsilon=\mp1\]

Euclidean

A Euclidean transformation is a geometric transformation that preserves the shape, size, and angles of an object. It includes rotation and translation but does not include scaling. It combines rotation, and translation into a single transformation.
A Euclidean transformation is a type of isometry
2D Euclidean Transformation:

Rotation: 1 DOF (rotation angle θ).

Translation: 2 DOF (translation in x and y directions, $t_x$ and $t_y$).

Total DOF in 2D: 1+2=3 degrees of freedom.

3D Euclidean Transformation:

Rotation: 3 DOF (rotation angles around the x, y, and z axes, often represented as Euler angles or a rotation matrix).

Translation: 3 DOF (translation in x, y, and z directions, $t_x$, $t_y$, and $t_z$).

Total DOF in 3D: 3+3=6 degrees of freedom.

Number of PC: 2
Invariants: All (length, angle, ratio of lengths, parallelism, includes cross ratio)

\[\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta & tx \\ \sin\theta & \cos\theta & ty \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}\]