Geometry

The study of shape, space, position, and measurement.

• Euclidean geometry:: flat space governed by five postulates, the classical framework
• non-Euclidean geometry:: curved spaces (hyperbolic, elliptic) where the parallel postulate fails
• differential geometry:: uses calculus on manifolds, the language of relativity
• Riemannian metric defines distance and curvature on smooth manifolds
• symmetry and transformation groups connect geometry to algebra
• projective geometry studies invariants under projection, central to computer vision
• fractal geometry describes self-similar structures with non-integer dimension
• Related: topology, linear algebra, graph theory, optimization, quantum mechanics