As a degenerate orbit, this is a radial elliptic trajectory. A complete orbit of this ellipse traverses the line segment twice. 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. The segment addition postulate can be used to add congruent segment or segments with equal lengths, and consequently substitute other segments into another statement to make segments congruent.Ī 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. This is important because it transforms some of the analysis of convex sets, to the analysis of a line segment. For example, in a convex set, the segment that joins any two points of the set is contained in the set. Segments play an important role in other theories. 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). 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. A pair of line segments can be any one of the following: intersecting, parallel, skew, or none of these.More generally than above, the concept of a line segment can be defined in an ordered geometry.
However, an open line segment is an open set in V if and only if V is one-dimensional. If V is a topological vector space, then a closed line segment is a closed set in V.A line segment is a connected, non-empty set.If V is a vector space over R Properties
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