Projective

Look at computer vision

Homogenous Coordinates

Make one more coordinate then you need. You decide the overall scale no longer matters. The point (1,3) is the same point as (2,6). The simple model is that of a pinhole camera. If the pinhole is at (0,0,0), then all point light sources on a line stretching through the pinhole make equivalent images. It strangely turns out that these coordinates have beautiful properties.

Duality

In homogenous coordinates,

Axiomatic

Euclidean

Classical geometry. Euclidean lines and points and circles

Algebraic

Differential

Visual differential geometry Functional differential geometry

Curvature

Discrete Geometry

Convex Geometry

See convex optimization