Rays on the real line. A ray on the real line is just an open set of the form (p,), or (-,p). A ray is also called a half line, or an open ray, to distinguish the notion of a closed ray, which includes its endpoint.

Properties Suppose p,q and pq.

  • (p,)(q,)=(q,).

  • (p,)(q,)=(p,).

  • (p,)(-,q)=(p,q) if pq, and if p=q.

  • (-,p)(q,)=.

  • (p,)(-,q)= if pq, and -{p} if p=q.

  • (-,p)(q,)=-[p,q] if pq, and -{p} if p=q.

Rays in a general Euclidean spaceMathworldPlanetmath. Let be a line in n and let p be a point lying on the . We may parameterize =(t) (parameter t) so that (0)=p. An (open) ray ρ lying on with endpoint p is the set of points


If the inequality t>0 is relaxed to t0 in the above expression, then we have a closed ray. Note that if the inequality above were changed to t<0 instead, we end up again with a ray lying on and endpoint p. It is a ray because we can reparameterize by using the parameter s=-t instead, so that


The difference between the two rays is that they point in the opposite directions. Therefore, in general, a ray can be characterized by

  • a line,

  • a point lying on the line, and

  • a direction on the line.

Rays in an ordered geometryMathworldPlanetmath: Given two distinct points p,q in an ordered geometry (A,B) (A is the underlying incidence geometry (http://planetmath.org/IncidenceGeometry) and B is the strict betweenness relation defined on the points of A). The set


where st¯ denotes the open line segment with endpoints s and t, is called the (open) ray generated by p and q emanating from p. It is denoted by pq. p in pq is called the source or the end point of the ray. A closed ray generated by p and q with endpoint p is the set pq{p}.


  • for any point rpq, pr=pq.

  • pqqp=pq and pqqp=pq¯. We say that a ray lies on a line if all of the points in the ray are incidentMathworldPlanetmath with the line. Also, a line segment lies on a ray if it is a subset of the ray.

  • The opposite ray of ρ=pq is defined to be


    It is denoted by -ρ.

  • The opposite ray -ρ of a ray ρ is ray. Suppose ρ=pq. Then ρ has the property that

    1. (a)

      ρ(-ρ)= and

    2. (b)


    Conversely, given a ray ρ=pq, any ray ρ satisfying the above two properties (replacing -ρ by ρ) is the opposite ray of ρ.

  • Given any point p on a line , there are exactly two rays lying on with endpoint p. Furthermore, p is between q and r in iff q and r lie on opposite rays on .

  • Given any two rays ρ and ϱ, exactly one of the following holds:

    1. (a)


    2. (b)

      ρϱ= a line segment, or

    3. (c)

      ρϱ= a ray.

    It is not hard to see that in the last case, one ray is included in the other, and their intersectionMathworldPlanetmath is the “smaller” of the two rays. In the first two cases, the two rays are said to be (pointing) in the opposite direction. In the last case, the two are said to be in the same direction. Opposite rays are clearly pointing in the opposite direction.

  • An equivalence relationMathworldPlanetmath can be defined on the set of all rays lying on a line by whether they are pointing in the same direction or not. Thus, the set of all rays lying on can be partitioned into two subsets R and R, so that if ρ,ϱR (or R), then they are pointing in the same direction; and if ρR and ϱR are pointing in the opposite direction.

  • Pick one of the two subsets from above, say R. Define on R by ρϱ if ρϱ. Then is a linear order on R. This induces a linear order on the line in the following way: pq if the corresponding rays ρ,ϱR, with endpoints p and q respectively, we have ρϱ. This is one way to define a linear ordering on a line . An alternative, but equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath way of defining a linear ordering on a line in an ordered geometry can be found in the entry under ordered geometry.

  • Note that in defining , we could have used R instead of R. This is an example of the duality of linear ordering.


  • 1 D. Hilbert, Foundations of Geometry, Open Court Publishing Co. (1971)
  • 2 K. Borsuk and W. Szmielew, Foundations of Geometry, North-Holland Publishing Co. Amsterdam (1960)
  • 3 M. J. Greenberg, EuclideanPlanetmathPlanetmath and Non-Euclidean Geometries, DevelopmentMathworldPlanetmath and History, W. H. Freeman and Company, San Francisco (1974)
Title ray
Canonical name Ray
Date of creation 2013-03-22 15:28:36
Last modified on 2013-03-22 15:28:36
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 51G05
Synonym half line
Synonym open ray
Related topic BetweennessInRays
Defines closed ray
Defines opposite ray