Module V·Article I·~1 min read
Groups of Transformations and Their Invariants
Affine and Projective Transformations
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Groups of Transformations
Klein's Erlangen Program
In 1872, Felix Klein proposed a revolutionary perspective on geometry: each geometry is determined by its group of transformations and studies the invariants of this group.
- Euclidean geometry: group of motions (isometries). Invariants: distance, angle, area.
- Affine geometry: group of affine transformations. Invariants: parallelism, ratio of lengths on parallel lines, area (up to scale).
- Projective geometry: group of projective transformations. Invariants: collinearity, cross ratio.
- Topology: group of homeomorphisms. Invariants: connectedness, compactness, number of holes.
Group of Motions
Motion = isometry: transformation preserving distances. In $\mathbb{R}^2$: rotations, reflections, parallel translations.
Proper motions (preserving orientation): rotations and translations. Form a subgroup. Improper (changing orientation): reflections and glide reflections.
Cross Ratio
For four points $A$, $B$, $C$, $D$ on a line: $(A,B;C,D) = (AC/BC)/(AD/BD)$.
Invariant under projective transformations. For special choices: harmonic division $(A,B;C,D) = -1$.
Homogeneous Coordinates
In $\mathbb{R}^n$, we use $\mathbb{R}P^n$ with homogeneous coordinates $(x_0:x_1:\ldots:x_n)$. Affine space: hyperplane $x_0=1$.
Transformations: linear over homogeneous coordinates $\rightarrow$ projective in affine space.
Parallel lines intersect at the point $x_0=0$ (at “infinity”). Perspective = projective transformation.
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