Two-dimensional space

These include analogs to physical spaces, like flat planes, and curved surfaces like spheres, cylinders, and cones, which can be infinite or finite.

On the Euclidean plane, any two points can be joined by a unique straight line along which the distance can be measured.

Two-dimensional spaces can also be curved, for example the sphere and hyperbolic plane, sufficiently small portions of which appear like the flat plane, but on which straight lines which are locally parallel do not stay equidistant from each-other but eventually converge or diverge, respectively.

Two-dimensional spaces with a locally Euclidean concept of distance but which can have non-uniform curvature are called Riemannian surfaces.

Lorentzian surfaces look locally like a two-dimensional slice of relativistic spacetime with one spatial and one time dimension; constant-curvature examples are the flat Lorentzian plane (a two-dimensional subspace of Minkowski space) and the curved de Sitter and anti-de Sitter planes.

A two-dimensional metric space has some concept of distance but it need not match the Euclidean version.

Vectors can be added together or scaled by a number, and optionally have a Euclidean, Lorentzian, or Galilean concept of distance.