ray
Rays on the real line. A ray on the real line is just an open set of the form , or . 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 and .
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if , and if .
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if , and if .
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if , and if .
Rays in a general Euclidean space. Let be a line in and let be a point lying on the . We may parameterize (parameter ) so that . An (open) ray lying on with endpoint is the set of points
If the inequality is relaxed to in the above expression, then we have a closed ray. Note that if the inequality above were changed to instead, we end up again with a ray lying on and endpoint . It is a ray because we can reparameterize by using the parameter 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
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a line,
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a point lying on the line, and
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a direction on the line.
Rays in an ordered geometry: Given two distinct points in an ordered geometry ( is the underlying incidence geometry (http://planetmath.org/IncidenceGeometry) and is the strict betweenness relation defined on the points of ). The set
where denotes the open line segment with endpoints and , is called the (open) ray generated by and emanating from . It is denoted by . in is called the source or the end point of the ray. A closed ray generated by and with endpoint is the set .
Properties.
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for any point , .
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and . We say that a ray lies on a line if all of the points in the ray are incident with the line. Also, a line segment lies on a ray if it is a subset of the ray.
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The opposite ray of is defined to be
It is denoted by .
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The opposite ray of a ray is ray. Suppose . Then has the property that
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and
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Conversely, given a ray , any ray satisfying the above two properties (replacing by ) is the opposite ray of .
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Given any point on a line , there are exactly two rays lying on with endpoint . Furthermore, is between and in iff and lie on opposite rays on .
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Given any two rays and , exactly one of the following holds:
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,
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a line segment, or
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a ray.
It is not hard to see that in the last case, one ray is included in the other, and their intersection 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.
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An equivalence relation 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 and , so that if (or ), then they are pointing in the same direction; and if and are pointing in the opposite direction.
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Pick one of the two subsets from above, say . Define on by if . Then is a linear order on . This induces a linear order on the line in the following way: if the corresponding rays , with endpoints and respectively, we have . This is one way to define a linear ordering on a line . An alternative, but equivalent way of defining a linear ordering on a line in an ordered geometry can be found in the entry under ordered geometry.
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Note that in defining , we could have used instead of . This is an example of the duality of linear ordering.
References
- 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, Euclidean and Non-Euclidean Geometries, Development and History, W. H. Freeman and Company, San Francisco (1974)
Title | ray |
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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 |