Line segment
The length of a line segment is given by the Euclidean distance between its endpoints.
In geometry, a line segment is often denoted using an overline (vinculum) above the symbols for the two endpoints, such as in AB.
[1] 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.
The endpoints of L are then the vectors u and u + v. Sometimes, one needs to distinguish between "open" and "closed" line segments.
Equivalently, a line segment is the convex hull of two points.
is the following collection of points: 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).
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.
In addition to appearing as the edges and diagonals of polygons and polyhedra, line segments also appear in numerous other locations relative to other geometric shapes.
Other segments of interest in a triangle include those connecting various triangle centers to each other, most notably the incenter, the circumcenter, the nine-point center, the centroid and the orthocenter.
Any straight line segment connecting two points on a circle or ellipse is called a chord.
Similarly, the shortest diameter of an ellipse is called the minor axis, and the segment from its midpoint (the ellipse's center) to either of its endpoints is called a semi-minor axis.
This suggestion has been absorbed into mathematical physics through the concept of a Euclidean vector.
[2][3] The collection of all directed line segments is usually reduced by making equipollent any pair having the same length and orientation.
[4] This application of an equivalence relation was introduced by Giusto Bellavitis in 1835.
