User:Boris Tsirelson/Sandbox1
Plane
Non-axiomatic approach
Definitions
A remark
To define a plane is more complicated than it may seem.
It is tempting to define a plane as a surface with zero curvature (or something like that). However, this is not a good idea, since the notions of surface and curvature are much more complicated than the notion of plane. In fact, several different notions of surface are introduced by topology and differential geometry, and several different notions of curvature are introduced by differential geometry; these are far beyond elementary mathematics. Fortunately, it is possible to define a plane via more elementary notions, and this way is preferred in mathematics. Still, some problems remain, see "axiomatic approach" below.
Several equivalent definitions of plane given below may be compared with the definition of Circle (mathematics) as consisting of those points in a plane that are a given distance (the radius) away from a given point (the center).
Definition via distances
Plane via right angles (orthogonality):
Plane via straight lines:
Plane via cartesian coordinates: