Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Line (geometry)
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
=== Ray ===<!-- This section is linked (older copy) from: [[Tangent]], [[Ray (mathematics)]], --> {{Redirect|Ray (geometry)|other uses in mathematics|Ray (disambiguation)#Science and mathematics}}[[File:Ray (A, B, C).svg|A ray with a terminus at A, with two points B and C on the right|alt=Ray|thumb]]{{See also|Orthant}} Given a line and any point ''A'' on it, we may consider ''A'' as decomposing this line into two parts. Each such part is called a '''ray''' and the point ''A'' is called its ''initial point''. It is also known as '''half-line''' (sometimes, a '''half-axis''' if it plays a distinct role, e.g., as part of a [[coordinate axis]]). It is a one-dimensional [[half-space (geometry)|half-space]]. The point A is considered to be a member of the ray.{{efn|On occasion we may consider a ray without its initial point. Such rays are called ''open'' rays, in contrast to the typical ray which would be said to be ''closed''.}} Intuitively, a ray consists of those points on a line passing through ''A'' and proceeding indefinitely, starting at ''A'', in one direction only along the line. However, in order to use this concept of a ray in proofs a more precise definition is required. Given distinct points ''A'' and ''B'', they determine a unique ray with initial point ''A''. As two points define a unique line, this ray consists of all the points between ''A'' and ''B'' (including ''A'' and ''B'') and all the points ''C'' on the line through ''A'' and ''B'' such that ''B'' is between ''A'' and ''C''.<ref>{{citation |last=Wylie Jr. |first=C.R. |title=Foundations of Geometry |year=1964 |at=p. 59, definition 3 |place=New York |publisher=McGraw-Hill |isbn=0-07-072191-2}}</ref> This is, at times, also expressed as the set of all points ''C'' on the line determined by ''A'' and ''B'' such that ''A'' is not between ''B'' and ''C''.<ref>{{citation |last=Pedoe |first=Dan |title=Geometry: A Comprehensive Course |page=2 |year=1988 |place=Mineola, NY |publisher=Dover |isbn=0-486-65812-0}}</ref> A point ''D'', on the line determined by ''A'' and ''B'' but not in the ray with initial point ''A'' determined by ''B'', will determine another ray with initial point ''A''. With respect to the ''AB'' ray, the ''AD'' ray is called the ''opposite ray''. Thus, we would say that two different points, ''A'' and ''B'', define a line and a decomposition of this line into the [[disjoint union]] of an open segment {{open-open|''A'',β''B''}} and two rays, ''BC'' and ''AD'' (the point ''D'' is not drawn in the diagram, but is to the left of ''A'' on the line ''AB''). These are not opposite rays since they have different initial points. In Euclidean geometry two rays with a common endpoint form an [[angle]].<ref>{{SpringerEOM|last=Sidorov|first=L. A.|date=2001|id=Angle&oldid=13323|title=Angle}}</ref> The definition of a ray depends upon the notion of betweenness for points on a line. It follows that rays exist only for geometries for which this notion exists, typically [[Euclidean geometry]] or [[affine geometry]] over an [[ordered field]]. On the other hand, rays do not exist in [[projective geometry]] nor in a geometry over a non-ordered field, like the [[complex number]]s or any [[finite field]].
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)