Editing Line (geometry) (section)

==Generalizations {{anchor|Generalizations}}==
{{see also|Geometric space}}

In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in [[analytic geometry]], a line in the plane is often defined as the set of points whose coordinates satisfy a given [[linear equation]], but in a more abstract setting, such as [[incidence geometry]], a line may be an independent object, distinct from the set of points which lie on it.

When a geometry is described by a set of [[axiom]]s, the notion of a line is usually left undefined (a so-called [[primitive notion|primitive]] object). The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in [[differential geometry]], a line may be interpreted as a [[geodesic]] (shortest path between points), while in some [[Projective geometry|projective geometries]], a line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line.

=== Projective geometry ===
{{Further|Geodesic}}
[[File:Great circle hemispheres.png|right|thumb|A great circle divides the sphere in two equal hemispheres, while also satisfying the "no curvature" property.]]
In many models of [[projective geometry]], the representation of a line rarely conforms to the notion of the "straight curve" as it is visualised in Euclidean geometry. In [[elliptic geometry]] we see a typical example of this.<ref name=":0" />{{Rp|page=108}} In the spherical representation of elliptic geometry, lines are represented by [[great circle]]s of a sphere with diametrically opposite points identified. In a different model of elliptic geometry, lines are represented by Euclidean [[plane (geometry)|planes]] passing through the origin. Even though these representations are visually distinct, they satisfy all the properties (such as, two points determining a unique line) that make them suitable representations for lines in this geometry.

The "shortness" and "straightness" of a line, interpreted as the property that the [[distance]] along the line between any two of its points is minimized (see [[triangle inequality]]), can be generalized and leads to the concept of [[geodesic]]s in [[metric space]]s.