Line segment

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The geometric definition of a closed line segment: the intersection of all points at or to the right of A with all points at or to the left of B Segment definition.svg
The geometric definition of a closed line segment: the intersection of all points at or to the right of A with all points at or to the left of B
historical image - create a line segment (1699) Fotothek df tg 0003359 Geometrie ^ Konstruktion ^ Strecke ^ Messinstrument.jpg
historical image – create a line segment (1699)

In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints (such as ). [1] [2]

Contents

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 . In which case, the vectors u and u + v are called the end points of L.

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 A = (ax, ay) and C = (cx, cy) is the following collection of points:

.

Properties

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, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets, to the analysis of a line segment. 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.

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 diagonals of polygons and polyhedra, line segments also appear in numerous other locations relative to other geometric shapes.

Triangles

Some very frequently considered segments in a triangle to include the three altitudes (each perpendicularly connecting a side or its extension to the opposite vertex), the three medians (each connecting a side's midpoint to the opposite vertex), the perpendicular bisectors of the sides (perpendicularly connecting the midpoint of a side to one of the other sides), and the internal angle bisectors (each connecting a vertex to the opposite side). In each case, there are various equalities relating these segment lengths to others (discussed in the articles on the various types of segment), as well as various inequalities.

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.

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 circle or ellipse is called a chord. Any chord in a circle which has no longer chord is called a diameter, and any segment connecting the circle's center (the midpoint of a diameter) to a point on the circle is called a radius.

In an ellipse, the longest chord, which is also the longest diameter, is called the major axis, and a segment from the midpoint of the major axis (the ellipse's center) to either endpoint of the major axis is called a semi-major axis. 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. The chords of an ellipse which are perpendicular to the major axis and pass through one of its foci are called the latera recta of the ellipse. The interfocal segment connects the two foci.

Directed line segment

When a line segment is given an orientation (direction) it suggests a translation or perhaps a force tending to make a translation. The magnitude and direction are indicative of a potential change. This suggestion has been absorbed into mathematical physics through the concept of a Euclidean vector. [3] [4] The collection of all directed line segments is usually reduced by making "equivalent" any pair having the same length and orientation. [5] This application of an equivalence relation dates from Giusto Bellavitis’s introduction of the concept of equipollence of directed line segments in 1835.

Generalizations

Analogous to straight line segments above, one can also define arcs as segments of a curve.

See also

Notes

  1. "List of Geometry and Trigonometry Symbols". Math Vault. 2020-04-17. Retrieved 2020-09-01.
  2. "Line Segment Definition - Math Open Reference". www.mathopenref.com. Retrieved 2020-09-01.
  3. Harry F. Davis & Arthur David Snider (1988) Introduction to Vector Analysis, 5th edition, page 1, Wm. C. Brown Publishers ISBN   0-697-06814-5
  4. Matiur Rahman & Isaac Mulolani (2001) Applied Vector Analysis, pages 9 & 10, CRC Press ISBN   0-8493-1088-1
  5. Eutiquio C. Young (1978) Vector and Tensor Analysis, pages 2 & 3, Marcel Dekker ISBN   0-8247-6671-7

Related Research Articles

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Ellipse Plane curve: conic section

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity , a number ranging from to .

Hyperbola Plane curve: conic section

In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

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Perpendicular Relationship between two lines that meet at a right angle (90 degrees)

In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects.

Parallelogram Quadrilateral with two pairs of parallel sides

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

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Midpoint

In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

Cone Geometric shape

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.

Line (geometry) Straight figure with zero width and depth

In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, which are often described in terms of two points or referred to using a single letter.

Lines in a plane or higher-dimensional space are said to be concurrent if they intersect at a single point.

Three-dimensional space Geometric model of the physical space

Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension.

Beltrami–Klein model

In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.

Conoid

In geometry a conoid is a ruled surface, whose rulings (lines) fulfill the additional conditions

Two-dimensional space Geometric model of the planar projection of the physical universe

Two-dimensional space is a geometric setting in which two values are required to determine the position of an element. The set 2 of pairs of real numbers with appropriate structure often serves as the canonical example of a two-dimensional Euclidean space. For a generalization of the concept, see dimension.

Pappus chain

In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD.

Conic section Curve obtained by intersecting a cone and a plane

In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.

Semi-major and semi-minor axes

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the widest points of the perimeter.

Poincaré disk model

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all circular arcs contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.

References

This article incorporates material from Line segment on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.