Coordinate Systems and Transformations

Affine Transformations

Affine transformation: $r' = Ar + b$ ($A$ is a non-singular matrix, $b$ is a vector).

Preserves: straightness, parallelism, ratios of lengths on parallel lines, area (with a coefficient $|\det A|$).

Examples: translation, rotation, scaling, shear, reflection.

The group of affine transformations of the plane: $3\times3$ matrices of the form $\begin{bmatrix}A & b \ 0 & 1\end{bmatrix}$ (homogeneous coordinates).

Rotation of Coordinates

When rotating the axes by an angle $\varphi$: $x = x'\cos\varphi - y'\sin\varphi$, $y = x'\sin\varphi + y'\cos\varphi$.

Reduction of second-order curves to canonical form: by rotation we eliminate the $xy$ term (the angle of rotation from $\tan(2\varphi) = \frac{B}{A-C}$).

Eigenvalues and Geometry

Matrix of the quadratic form $A = \begin{bmatrix}A & B/2 \ B/2 & C\end{bmatrix}$. Eigenvalues $\lambda_1$, $\lambda_2$ — coefficients of the canonical form $\lambda_1 x^2 + \lambda_2 y^2$. The type of curve is determined by the signs of $\lambda_1$, $\lambda_2$.

Projective Transformations

Projective transformation: $r' \sim Pr$ ($P$ is a non-singular $3\times3$ matrix in homogeneous coordinates).

Invariants: cross-ratio of four points on a line. In computer vision: perspective transformations are projective.