euclidian geometry

Euclidian Geometry

Euclidean geometry is the study of invariants under rigid motions (isometries).
In other words: Properties that stay true when you translate, rotate, or reflect objects, but not when you stretch or bend them.

Several equivalent ways to define it

Axiomatic approach

Euclidean geometry is the mathematical theory derived from Euclid’s five postulates (or more rigorously, Hilbert’s axioms). Key axioms include:

• Points and lines exist with incidence relations
• Unique line through two points
• Angles and distances can be compared
• Parallel postulate: Through a point not on a line, there exists exactly one parallel line

Transformation group approach (Klein’s Erlangen program)

Euclidean geometry is the study of properties invariant under the Euclidean group $E(n)$, which consists of:

• Translations
• Rotations
• Reflections

These are the isometries of ${\mathbb{R}}^n$ - transformations that preserve distance: $d(T(\mathbf{x}),T(\mathbf{y}))=d(\mathbf{x},\mathbf{y})$.

Metric geometry approach

Euclidean geometry is geometry in a metric space $({\mathbb{R}}^n,d)$ where $d$ is the Euclidean metric. It studies:

• Distances and angles
• Congruence (figures related by isometries)
• Concepts like lines (geodesics), circles, triangles
• Properties like the Pythagorean theorem, angle sum in triangles = $180°$