Line segment
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Geometers |
In
Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. When the end points both lie on a curve (such as a circle), a line segment is called a chord (of that curve).
In real or complex vector spaces
If V is a vector space over or and L is a subset of V, then L is a line segment if L can be parameterized as
for some vectors where v is nonzero. The endpoints of L are then the vectors u and u + v.
Sometimes, one needs to distinguish between "open" and "closed" line segments. In this case, one would define a closed line segment as above, and an open line segment as a subset L that can be parametrized as
for some vectors
Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points.
In geometry, one might define point B to be between two other points A and C, if the distance |AB| added to the distance |BC| is equal to the distance |AC|. Thus in the line segment with endpoints and is the following collection of points:
Properties
- A line segment is a non-empty set.
- If V is a .
- More generally than above, the concept of a line segment can be defined in an ordered geometry.
- A pair of line segments can be any one of the following: intersecting, parallel, skew, or none of these. The last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane then they must cross each other, but that need not be true of segments.
In proofs
In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or defined in terms of an isometry of a line (used as a coordinate system).
Segments play an important role in other theories. For example, in a
As a degenerate ellipse
A line segment can be viewed as a degenerate case of an ellipse, in which the semiminor axis goes to zero, the foci go to the endpoints, and the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant; if this constant equals the distance between the foci, the line segment is the result. A complete orbit of this ellipse traverses the line segment twice. As a degenerate orbit, this is a radial elliptic trajectory.
In other geometric shapes
In addition to appearing as the edges and
Triangles
Some very frequently considered segments in a
Other segments of interest in a triangle include those connecting various
Quadrilaterals
In addition to the sides and diagonals of a quadrilateral, some important segments are the two bimedians (connecting the midpoints of opposite sides) and the four maltitudes (each perpendicularly connecting one side to the midpoint of the opposite side).
Circles and ellipses
Any straight line segment connecting two points on a
In an ellipse, the longest chord, which is also the longest
Directed line segment
When a line segment is given an
Generalizations
Analogous to
In one-dimensional space, a ball is a line segment.
An
Types of line segments
See also
- Polygonal chain
- Interval (mathematics)
- Line segment intersection, the algorithmic problem of finding intersecting pairs in a collection of line segments
Notes
- ^ "Line Segment Definition - Math Open Reference". www.mathopenref.com. Retrieved 2020-09-01.
- ISBN 0-697-06814-5
- ISBN 0-8493-1088-1
- ISBN 0-8247-6671-7
References
- David Hilbert The Foundations of Geometry. The Open Court Publishing Company 1950, p. 4
External links
- Weisstein, Eric W. "Line segment". MathWorld.
- Line Segment at PlanetMath
- Copying a line segment with compass and straightedge
- Dividing a line segment into N equal parts with compass and straightedge Animated demonstration
This article incorporates material from Line segment on