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Line segment

In Euclidean geometry, a line segment is a finite portion of a straight line bounded by two distinct endpoints, including all points between them along the line.^[1] It is typically denoted by the endpoints, such as \overline{AB}, where A and B are the endpoints, and it represents the shortest path connecting those points in a plane or space.^[2] Unlike an infinite line, a line segment has a defined length and no extension beyond its bounds.^[1] Line segments form the foundational building blocks of geometric constructions, serving as the sides of polygons, the edges of polyhedra, and the basis for measuring distances and angles.^[3] The length of a line segment between two points (x_1, y_1) and (x_2, y_2) in the Euclidean plane is given by the distance formula \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, which quantifies the straight-line separation.^[4] Key properties include uniqueness—only one line segment exists between any two distinct points.^[2] It can be bisected at its midpoint, dividing it into two congruent segments.^[5] Additionally, line segments exhibit congruence if they have equal lengths, a principle central to proofs in geometry. In higher mathematics, line segments extend to vector spaces, where they are parameterized as convex combinations of endpoints, such as (1-t)A + tB for $0 \leq t \leq 1, preserving properties like connectedness and closure.^[1] They play roles in computational geometry for algorithms involving intersections and in Euclidean postulates, such as the ability to draw a straight line segment between any two points.^[6]

Fundamentals

Definition

In mathematics, a line segment is defined as a finite portion of a straight line bounded by two distinct endpoints, forming a closed interval that includes all points between and including those endpoints.^[1] More formally, it is the convex hull of two distinct points in Euclidean space, consisting of the set of all points that can be expressed as a convex combination of the endpoints.^[7] This makes it the shortest path connecting the two points along the straight line in a metric space like the Euclidean plane or space.^[1] Unlike an infinite line, which extends without bound in both directions and has no endpoints, a line segment is limited to the region between its specified endpoints.^[8] Similarly, a ray differs by starting at one endpoint and extending infinitely in only one direction, lacking a second bounding point.^[9] These distinctions highlight the line segment's bounded and complete nature, encompassing both endpoints and the path between them. The concept of a line segment originates in Euclidean geometry, where it was formalized around 300 BCE in Euclid's Elements through postulates describing straight lines, such as the assertion that a straight line segment can be drawn joining any two points.^[6] For example, in the Cartesian plane, the line segment AB connects point A at (0,0) to point B at (1,1), including all points on the straight path between them.^[1]

Notation and Representation

In Euclidean geometry, a line segment connecting two points A and B is commonly denoted using an overline symbol, written as \overline{AB}, which indicates the finite portion of the line between the endpoints.^[10] This notation emphasizes the segment's bounded nature and is standard in synthetic geometry texts.^[11] For directed segments in vector contexts, alternative representations include boldface letters, such as \mathbf{AB}, or an arrow above the points, \overrightarrow{AB}, to convey direction from A to B.^[12]^[13] In the Cartesian plane, a line segment between points with position vectors \mathbf{A} and \mathbf{B} is represented parametrically as the set \{ (1-t)\mathbf{A} + t\mathbf{B} \mid t \in [0,1] \}, where t is a parameter ranging from 0 to 1, capturing all points along the segment.^[14] This form highlights the convex combination of the endpoints and is prevalent in analytic geometry.^[15] Visually, line segments are depicted in diagrams as straight lines with distinct marks at the endpoints A and B to identify them clearly.^[16] To indicate congruence between segments, hash marks (short perpendicular lines) are often added along the segments, with matching numbers or styles of marks denoting equal lengths.^[17]^[18] Notation varies by context: synthetic geometry favors the overline \overline{AB} for its simplicity in proofs, while analytic geometry on the real line treats the segment as a closed interval [a, b], where a and b are the coordinates of the endpoints assuming a \leq b.^[19] This interval notation aligns with set-theoretic representations in analysis.^[19]

Properties

Length and Midpoint

The length of a line segment in Euclidean space is defined as the Euclidean distance between its endpoints, providing a measure of the shortest path connecting them. For two points A = (a_1, a_2, \dots, a_n) and B = (b_1, b_2, \dots, b_n) in n-dimensional space, the length d(A, B) is calculated using the formula

d(A, B) = \sqrt{\sum_{k=1}^n (b_k - a_k)^2}.

This formula derives from the Pythagorean theorem generalized to higher dimensions, where the length represents the straight-line distance along the segment.^[4]^[20] The length is a positive scalar value for distinct endpoints, reflecting the segment's magnitude, and it equals zero only when the endpoints coincide, indicating a degenerate segment. This property ensures the length serves as a reliable metric in geometric constructions. Additionally, the length remains invariant under rigid transformations—such as translations, rotations, and reflections—which preserve distances between points without altering the segment's measure.^[4]^[21] The midpoint of a line segment is the point that divides it into two equal subsegments, located at the average of the endpoints' coordinates. For endpoints A = (a_1, a_2, \dots, a_n) and B = (b_1, b_2, \dots, b_n), the midpoint M is given by

M = \left( \frac{a_1 + b_1}{2}, \frac{a_2 + b_2}{2}, \dots, \frac{a_n + b_n}{2} \right).

This position ensures each subsegment from A to M or M to B has length d(A, B)/2, maintaining symmetry along the segment.^[22]^[23] For example, consider a line segment AB with A = (0, 0) and B = (4, 0) in the plane. The length is d(A, B) = \sqrt{(4-0)^2 + (0-0)^2} = 4, and the midpoint is M = \left( \frac{0+4}{2}, \frac{0+0}{2} \right) = (2, 0), dividing the segment into two parts of length 2 each.^[4]^[22]

Parameterization and Division

A line segment connecting two points A and B in Euclidean space can be parameterized using a scalar parameter t that varies from 0 to 1, where the position vector of a point P(t) on the segment is given by

\mathbf{P}(t) = \mathbf{A} + t(\mathbf{B} - \mathbf{A}), \quad t \in [0, 1].

This formulation, known as linear interpolation, traces the segment continuously from A (when t=0) to B (when t=1), providing a convenient way to describe all points along the segment.^[14] The parameterization facilitates the division of the segment in specified ratios. For internal division, where a point P lies between A and B and divides the segment in the ratio m:n (with m and n positive integers), the coordinates of P are calculated using the section formula:

\mathbf{P} = \frac{n\mathbf{A} + m\mathbf{B}}{m + n}.

This formula weights the positions of A and B according to the ratio, ensuring P is closer to the endpoint with the larger weight. The midpoint, which divides the segment in the ratio 1:1, is a special case yielding \mathbf{P} = \frac{\mathbf{A} + \mathbf{B}}{2}.^[24]^[25] External division occurs when the dividing point P lies outside the segment, extending the line beyond one endpoint in the ratio m:n (where m and n may have opposite signs to indicate direction). The corresponding section formula is

\mathbf{P} = \frac{n\mathbf{A} - m\mathbf{B}}{n - m}.

This extends the parameterization beyond the interval t \in [0, 1], allowing t < 0 or t > 1 to locate points on the infinite line but outside the bounded segment.^[26]^[27] In computer graphics, the parametric form of line segments is essential for linear interpolation, enabling the rendering of straight lines by incrementally computing pixel positions between endpoints. This technique underpins algorithms like Bresenham's line algorithm and supports smooth transitions in animations and shading models.^[28]^[29]

Applications in Euclidean Geometry

In Polygons

In polygons, line segments serve as the fundamental building blocks, forming the sides that bound the shape and the diagonals that connect non-adjacent vertices. In a triangle, the three sides are line segments that enclose the interior and constitute the perimeter, with the semiperimeter defined as half the sum of these lengths.^[30] Internal line segments such as medians—connecting each vertex to the midpoint of the opposite side—and altitudes—perpendicular segments from each vertex to the line containing the opposite side—play key roles in dividing the triangle and determining properties like area and balance points.^[30] These medians intersect at the centroid, which divides each median in a 2:1 ratio,^[31] while the altitudes concur at the orthocenter.^[32] In quadrilaterals, four line segments form the sides, with two additional diagonals connecting opposite vertices to divide the shape into triangles. A notable property involves the midpoints of these sides: joining them creates a parallelogram, known as the Varignon parallelogram, regardless of the quadrilateral's convexity, as established by Pierre Varignon in 1731.^[33] The sides of this resulting parallelogram are parallel to the diagonals of the original quadrilateral and half their length, highlighting the relational geometry of line segments in such figures.^[34] For any polygon with n sides, the side lengths must satisfy the polygon inequality: the sum of the lengths of any n-1 sides exceeds the length of the remaining side, ensuring the segments can close to form a bounded region without degeneracy.^[35] This condition generalizes the triangle inequality and is necessary for the existence of a simple polygon in the plane.^[35] In regular polygons, all sides are congruent line segments of equal length, arranged with rotational symmetry around the center.^[36] The diagonals, also line segments, vary in length based on the number of vertices skipped; for example, in a regular pentagon, shorter diagonals skip one vertex while longer ones skip two, creating a hierarchy of segment lengths that contribute to the polygon's symmetry and tiling properties.^[36] The congruence of sides in these polygons ensures equal perimeter contributions from each segment.

In Conic Sections

In conic sections, a line segment plays a key role as a chord, which is the straight line connecting two points on the curve. For circles, a special case of conics, the perpendicular from the center to a chord bisects the chord, dividing it into two equal segments.^[37] This property follows from the symmetry of the circle and the equal radii to the chord's endpoints. Additionally, the length of such a chord is given by the formula $2\sqrt{r^2 - d^2}, where r is the radius of the circle and d is the perpendicular distance from the center to the chord.^[38] In ellipses, line segments appear as focal chords, which are chords passing through one of the foci. These chords are significant in describing the ellipse's geometry and parametric properties. A particular focal chord is the latus rectum, defined as the chord through a focus perpendicular to the major axis; its length is \frac{2b^2}{a}, where a and b are the semi-major and semi-minor axes, respectively.^[39] This segment helps in deriving the ellipse's polar equation and understanding focal distances.^[40] A degenerate case occurs when the semi-minor axis b approaches zero, causing the ellipse to collapse into a line segment of length $2a along the major axis.^[41] This limit illustrates how conic sections can reduce to simpler geometric figures under specific parameter conditions. Tangent line segments from an external point to a circle are equal in length, a theorem arising from congruent right triangles formed by the radii to the points of tangency and the external point.^[42] This equality holds because the tangents are perpendicular to the radii at the contact points, ensuring symmetric distances from the external point.

Vector and Directed Segments

In Vector Spaces

In a real vector space V, a line segment between two points A, B \in V is defined as the convex hull of \{A, B\}, consisting of all points of the form (1-t)A + tB where t \in [0,1].^[43]^[44] This representation arises from affine combinations, where the coefficients are non-negative and sum to 1, distinguishing it from general linear combinations that may extend beyond the segment.^[45] The parameterization aligns with the basic form used in Euclidean settings, but here it generalizes to any dimension without assuming an inner product.^[46] In the complex plane, viewed as a two-dimensional real vector space or one-dimensional complex vector space, a line segment connects two complex numbers a and b via the set \{(1-t)a + tb \mid t \in [0,1]\}.^[47] These points are plotted on the Argand diagram, where the real part corresponds to the horizontal axis and the imaginary part to the vertical axis, illustrating the segment as a straight path in the plane.^[48] This visualization emphasizes the vector space structure, treating complex numbers as vectors from the origin. Equipped with a norm \|\cdot\|, the line segment inherits topological properties from the normed space: it is closed as the continuous image of the compact interval [0,1], convex by construction since it contains all affine combinations of its endpoints, and compact in finite-dimensional spaces due to the Heine-Borel theorem, as it is closed and bounded.^[44]^[49] The metric d(x,y) = \|x - y\| induces a length on the segment equal to \|B - A\|, and the segment itself serves as a geodesic path connecting A and B.^[50] While the affine definition of the line segment remains consistent across norms, the geometry of "straightness" varies: in the Euclidean norm (\ell_2), the segment is the unique shortest path, but in general norms like the Manhattan norm (\ell_1), multiple paths of equal length exist between endpoints, though the affine segment is still a geodesic.^[50]^[51] The Euclidean norm is typically assumed as standard for its unique geodesic property in finite dimensions.^[51]

Directed Line Segments

A directed line segment is defined as an ordered pair of distinct points A and B in a Euclidean space, with an inherent orientation from the initial point A to the terminal point B.^[52] This orientation distinguishes it from an undirected line segment, which treats the endpoints symmetrically without specifying direction.^[13] Mathematically, the directed line segment is represented by the vector \overrightarrow{AB} = B - A, where A and B are position vectors relative to some origin, capturing both the displacement and direction from A to B.^[53] The length of a directed line segment \overrightarrow{AB} is the magnitude of the vector, given by \|\overrightarrow{AB}\| = \sqrt{(B_x - A_x)^2 + (B_y - A_y)^2 + \cdots}, which remains positive regardless of orientation.^[13] In one dimension, however, the length can be considered signed, reflecting the direction: positive if B > A and negative if B < A, allowing for algebraic operations along a number line.^[54] This signed interpretation facilitates computations in oriented contexts, such as determining relative positions on a line. Operations on directed line segments follow vector arithmetic. Addition of two directed segments \overrightarrow{AB} and \overrightarrow{PQ} adheres to the parallelogram law: the resultant segment starts at A and ends at a point R such that ABQR forms a parallelogram, equivalent to \overrightarrow{AB} + \overrightarrow{PQ} = \overrightarrow{AR}.^[55] Scalar multiplication by a real number k scales the segment: if k > 0, it preserves direction and stretches by factor |k|; if k < 0, it reverses direction while scaling.^[56] These operations enable composition of displacements and transformations in geometric constructions. In physics, directed line segments model displacement vectors, representing the change in position of an object from initial point A to final point B, independent of the path taken.^[52] For instance, the displacement \overrightarrow{AB} quantifies straight-line motion with both magnitude (distance traveled) and direction (heading). In geometry, they underpin oriented angles: the angle between two directed segments sharing an initial point is measured positively in the counterclockwise direction from the first to the second, enabling signed measures for rotations and turns in figures like polygons.^[54]

Advanced and Generalized Concepts

Generalizations to Higher Dimensions

In n-dimensional Euclidean space, a line segment is defined as the convex hull of two distinct points, serving as the fundamental 1-dimensional building block in higher-dimensional geometric structures. Specifically, it corresponds to the 1-simplex, which is the edge connecting two vertices in an n-simplex—the convex hull of n+1 affinely independent points.^[57] In the context of polytopes, which are bounded convex polyhedra generalized to n dimensions, line segments form the 1-dimensional faces known as edges, linking pairs of vertices while satisfying the polytope's convexity.^[58] For instance, in 3-dimensional space, these edges appear as the line segments bounding the faces of polyhedra like the octahedron (with 12 edges) or dodecahedron (with 30 edges), and more generally, line segments extend to space diagonals that connect non-adjacent vertices within the polytope, providing internal straight paths.^[58] Beyond Euclidean spaces, line segments generalize to geodesic segments in non-Euclidean geometries, representing the shortest paths between two points on curved manifolds. On a sphere, these geodesics are arcs of great circles, which are the intersections of the sphere with planes passing through its center, minimizing distance unlike straight lines in flat space.^[59] In hyperbolic geometry, such as the hyperbolic plane modeled by the upper half-plane \{z = x + iy \mid y > 0\} with metric ds^2 = (dx^2 + dy^2)/y^2, geodesics include vertical lines perpendicular to the real axis and semicircles orthogonal to it, both serving as locally distance-minimizing curves that replace Euclidean straight segments.^[60] These geodesic segments capture the intrinsic geometry of the space, where parallel lines diverge and triangle angle sums deviate from 180 degrees. In abstract metric spaces, the concept further extends to any curve that locally realizes the shortest path between two points, without requiring an embedding in Euclidean space. A geodesic in a length metric space (X, d) is a path \gamma: [a, b] \to X such that for every subinterval, its length equals the metric distance between endpoints, generalizing the straight line segment as the "straightest" possible connection under the given metric.^[61] This notion applies to curved domains, where geodesics may unfold as straight line segments in local parameterizations but globally curve to minimize total length.^[62] In computational geometry, line segments in higher dimensions underpin algorithms for constructing convex hulls, the minimal convex set containing a point cloud, which in vector spaces includes all line segments between points as per the convex combination definition. For example, in 2D, the Graham scan algorithm sorts points by polar angle around a base point and iteratively tests line segment orientations using cross-product checks to ensure counterclockwise turns, eliminating interior points to form the hull boundary composed of these segments, achieving O(n log n) time complexity; in higher dimensions, algorithms such as gift wrapping are used.^[63]^[64]

Degenerate and Special Cases

A line segment arises as a degenerate conic section when an ellipse collapses under limiting conditions. Specifically, the standard equation of an ellipse, \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, degenerates into the line segment along the major axis from (-a, 0) to (a, 0) as the semi-minor axis b approaches 0, effectively flattening the curve into a one-dimensional object with zero thickness. This degeneration highlights how higher-dimensional curves can reduce to lower-dimensional primitives in algebraic geometry.^[65] In Euclidean geometry proofs and axioms, line segments serve as fundamental building blocks. The Side-Angle-Side (SAS) congruence postulate relies on the equality of line segments for two sides and the included angle to establish triangle congruence, forming a core axiom for rigid body transformations.^[66] Degenerate triangles, where three collinear points form a "triangle" with zero area, essentially reduce to a single line segment connecting the outermost points, illustrating boundary cases in triangle inequality theorems where the sum of two sides equals the third.^[67] Special cases of line segments include zero-length instances, which degenerate into a single point, as the endpoints coincide and the distance metric yields zero.^[68] Infinite extensions encompass rays (extending infinitely in one direction from an endpoint) and full lines (extending infinitely in both directions), broadening the finite segment concept in unbounded geometric contexts.^[69] In projective geometry, line segments incorporate points at infinity, where parallel segments are idealized to intersect on the line at infinity, unifying affine and infinite perspectives in a single plane.^[70] This framework treats the line at infinity as a projective line comprising ideal points, enabling consistent handling of directions without endpoints.^[71]

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A closed interval corresponding to a finite portion of an infinite line. Line segments are generally labeled with two letters corresponding to their endpoints.
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[PDF] A Survey of Algorithms for Geodesic Paths and Distances
Jul 20, 2020 · This survey reviews algorithms for computing geodesic paths and distances on curved domains, which are paths through curved domains.
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[PDF] 1 Convex Hulls - Jeff Erickson
The convex hull is the smallest convex polygon containing the points, where for any two points inside, the entire line segment lies inside.<|control11|><|separator|>
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Convex Hull - Algorithms, 4th Edition
A convex hull is the minimum area convex region containing all points in a set. A set is convex if any two points' line segment is inside the set.
[61]
Degenerate Conics II: Are Their Parameters Meaningful?
Feb 25, 2022 · Here we'll look at the parameters of conic sections (focus, directrix, axes, and especially eccentricity) and how they apply to degenerate cases.
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Degenerate Conics I: Mystery of the Missing Case - The Math Doctors
Feb 18, 2022 · The general conic has an equation of the form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. If A = B = C = 0, then the cone degenerates to a line. If all ...
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[PDF] Postulates and Theorems - Geometry - Fort Bend ISD
Side-Angle-Side (SAS) Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then ...
[64]
Degenerate - Mathwords
A degenerate triangle is the "triangle" formed by three collinear points. It doesn't look like a triangle, it looks like a line segment. A parabola may be ...
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Can a triangle have a side length of zero? - Math Stack Exchange
Feb 17, 2017 · Yes, and any such triangle is called a degenerate triangle. If the distance between two points a and b is zero, then the corresponding vertex ¯ ...How do you call a segment of length 0? Does it have a name?How can points that have length zero result in a line segment with ...More results from math.stackexchange.com
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Lines, Segments, and Rays - Varsity Tutors
A line segment has two endpoints, and a ray starts at one endpoint and extends infinitely in one direction. Important Notes. •A line is represented by arrows at ...
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Projective Infinity - Interactive Mathematics Miscellany and Puzzles
Addition of the points and the line at infinity metamorphoses the Euclidean plane into the projective plane and Desargues was one of the founders of projective ...
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[PDF] Projective Geometry - Purdue Computer Science
▶ The projective plane adds points at infinity to the affine plane. ▶ Two parallel lines intersect at a point at infinity.