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In geometry, collinearity of a set of
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
is the property of their lying on a single
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".


Points on a line

In any geometry, the set of points on a line are said to be collinear. In Euclidean geometry this relation is intuitively visualized by points lying in a row on a "straight line". However, in most geometries (including Euclidean) a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
is typically a primitive (undefined) object type, so such visualizations will not necessarily be appropriate. A model for the geometry offers an interpretation of how the points, lines and other object types relate to one another and a notion such as collinearity must be interpreted within the context of that model. For instance, in
spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...
, where lines are represented in the standard model by great circles of a sphere, sets of collinear points lie on the same great circle. Such points do not lie on a "straight line" in the Euclidean sense, and are not thought of as being ''in a row''. A mapping of a geometry to itself which sends lines to lines is called a collineation; it preserves the collinearity property. The linear maps (or linear functions) of vector spaces, viewed as geometric maps, map lines to lines; that is, they map collinear point sets to collinear point sets and so, are collineations. In projective geometry these linear mappings are called '' homographies'' and are just one type of collineation.


Examples in Euclidean geometry


Triangles

In any triangle the following sets of points are collinear: *The orthocenter, the circumcenter, the centroid, the
Exeter point In geometry, the Exeter point is a special point associated with a plane triangle. The Exeter point is a triangle center and is designated as the center X(22) in Clark Kimberling's Encyclopedia of Triangle Centers. This was discovered in a comput ...
, the
de Longchamps point In geometry, the de Longchamps point of a triangle is a triangle center named after French mathematician Gaston Albert Gohierre de Longchamps. It is the reflection of the orthocenter of the triangle about the circumcenter.. Definition Let the ...
, and the center of the nine-point circle are collinear, all falling on a line called the Euler line. *The de Longchamps point also has other collinearities. *Any vertex, the tangency of the opposite side with an excircle, and the Nagel point are collinear in a line called a splitter of the triangle. *The midpoint of any side, the point that is equidistant from it along the triangle's boundary in either direction (so these two points bisect the perimeter), and the center of the Spieker circle are collinear in a line called a cleaver of the triangle. (The Spieker circle is the incircle of the medial triangle, and its center is the
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
of the perimeter of the triangle.) *Any vertex, the tangency of the opposite side with the incircle, and the Gergonne point are collinear. *From any point on the circumcircle of a triangle, the nearest points on each of the three extended sides of the triangle are collinear in the Simson line of the point on the circumcircle. *The lines connecting the feet of the altitudes intersect the opposite sides at collinear points.Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929). *A triangle's incenter, the midpoint of an altitude, and the point of contact of the corresponding side with the excircle relative to that side are collinear.Altshiller-Court, Nathan. ''College Geometry'', Dover Publications, 1980. * Menelaus' theorem states that three points P_1, P_2, P_3 on the sides (some
extended Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Exte ...
) of a triangle opposite vertices A_1,A_2, A_3 respectively are collinear if and only if the following products of segment lengths are equal: ::P_1A_2 \cdot P_2A_3 \cdot P_3A_1=P_1A_3 \cdot P_2A_1 \cdot P_3A_2. * The incenter, the centroid, and the Spieker circle's center are collinear. *The circumcenter, the Brocard midpoint, and the
Lemoine point In geometry, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians (medians reflected at the associated angle bisectors) of a triangle. Ross Honsberger called its existence "one of the crown jewels of m ...
of a triangle are collinear. *Two
perpendicular lines In elementary geometry, two geometric objects are perpendicular if they intersection, intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ...
intersecting at the orthocenter of a triangle each intersect each of the triangle's
extended side In plane geometry, an extended side or sideline of a polygon is the line that contains one side of the polygon. The extension of a side arises in various contexts. Triangle In an obtuse triangle, the altitudes from the acute angled vertices i ...
s. The midpoints on the three sides of these points of intersection are collinear in the Droz–Farny line.


Quadrilaterals

*In a convex quadrilateral whose opposite sides intersect at and , the midpoints of are collinear and the line through them is called the Newton line (sometimes known as the Newton-Gauss line). If the quadrilateral is a
tangential quadrilateral In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the ...
, then its incenter also lies on this line. *In a convex quadrilateral, the quasiorthocenter , the "area centroid" , and the quasicircumcenter are collinear in this order, and .. (See Quadrilateral#Remarkable points and lines in a convex quadrilateral.) *Other collinearities of a
tangential quadrilateral In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the ...
are given in Tangential quadrilateral#Collinear points. *In a cyclic quadrilateral, the circumcenter, the vertex centroid (the intersection of the two bimedians), and the anticenter are collinear. *In a cyclic quadrilateral, the area centroid, the vertex centroid, and the intersection of the diagonals are collinear. *In a tangential trapezoid, the tangencies of the incircle with the two bases are collinear with the incenter. *In a tangential trapezoid, the midpoints of the legs are collinear with the incenter.


Hexagons

* Pascal's theorem (also known as the Hexagrammum Mysticum Theorem) states that if an arbitrary six points are chosen on a conic section (i.e.,
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon. The converse is also true: the Braikenridge–Maclaurin theorem states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line, then the six vertices of the hexagon lie on a conic, which may be degenerate as in Pappus's hexagon theorem.


Conic sections

*By
Monge's theorem In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinea ...
, for any three circles in a plane, none of which is completely inside one of the others, the three intersection points of the three pairs of lines, each externally tangent to two of the circles, are collinear. *In an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
, the center, the two
foci Focus, or its plural form foci may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film *''Focus'', a 1962 TV film starring James Whitmore * ''Focus'' (2001 film), a 2001 film based ...
, and the two vertices with the smallest radius of curvature are collinear, and the center and the two vertices with the greatest radius of curvature are collinear. *In a hyperbola, the center, the two foci, and the two vertices are collinear.


Cones

*The
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
of a
conic solid A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines co ...
of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.


Tetrahedrons

*The centroid of a tetrahedron is the midpoint between its
Monge point In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
and circumcenter. These points define the ''Euler line'' of the tetrahedron that is analogous to the Euler line of a triangle. The center of the tetrahedron's twelve-point sphere also lies on the Euler line.


Algebra


Collinearity of points whose coordinates are given

In coordinate geometry, in -dimensional space, a set of three or more distinct points are collinear if and only if, the matrix of the coordinates of these vectors is of rank 1 or less. For example, given three points :\begin X &= (x_1,\ x_2,\ \dots,\ x_n), \\ Y &= (y_1,\ y_2,\ \dots,\ y_n), \\ Z &= (z_1,\ z_2,\ \dots,\ z_n), \end if the
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
:\begin x_1 & x_2 & \dots & x_n \\ y_1 & y_2 & \dots & y_n \\ z_1 & z_2 & \dots & z_n \end is of rank 1 or less, the points are collinear. Equivalently, for every subset of , if the
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
:\begin 1 & x_1 & x_2 & \dots & x_n \\ 1 & y_1 & y_2 & \dots & y_n \\ 1 & z_1 & z_2 & \dots & z_n \end is of rank 2 or less, the points are collinear. In particular, for three points in the plane (), the above matrix is square and the points are collinear if and only if its determinant is zero; since that 3 × 3 determinant is plus or minus twice the area of a triangle with those three points as vertices, this is equivalent to the statement that the three points are collinear if and only if the triangle with those points as vertices has zero area.


Collinearity of points whose pairwise distances are given

A set of at least three distinct points is called straight, meaning all the points are collinear, if and only if, for every three of those points , the following determinant of a
Cayley–Menger determinant In linear algebra, geometry, and trigonometry, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional volume, of a n-dimensional simplex in terms of the squares of all of the distances between pairs of its v ...
is zero (with meaning the distance between and , etc.): :: \det \begin 0 & d(AB)^2 & d(AC)^2 & 1 \\ d(AB)^2 & 0 & d(BC)^2 & 1 \\ d(AC)^2 & d(BC)^2 & 0 & 1 \\ 1 & 1 & 1 & 0 \end = 0. This determinant is, by Heron's formula, equal to −16 times the square of the area of a triangle with side lengths ; so checking if this determinant equals zero is equivalent to checking whether the triangle with vertices has zero area (so the vertices are collinear). Equivalently, a set of at least three distinct points are collinear if and only if, for every three of those points with greater than or equal to each of and , the triangle inequality holds with equality.


Number theory

Two numbers and are not coprime—that is, they share a common factor other than 1—if and only if for a rectangle plotted on a
square lattice In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as . It is one of the five types of two-dimensional lattices as classified by their ...
with vertices at , at least one interior point is collinear with and .


Concurrency (plane dual)

In various plane geometries the notion of interchanging the roles of "points" and "lines" while preserving the relationship between them is called plane duality. Given a set of collinear points, by plane duality we obtain a set of lines all of which meet at a common point. The property that this set of lines has (meeting at a common point) is called concurrency, and the lines are said to be concurrent lines. Thus, concurrency is the plane dual notion to collinearity.


Collinearity graph

Given a
partial geometry An incidence structure C=(P,L,I) consists of points P, lines L, and flags I \subseteq P \times L where a point p is said to be incident with a line l if (p,l) \in I. It is a (finite) partial geometry if there are integers s,t,\alpha\geq 1 such that: ...
, where two points determine at most one line, a collinearity graph of is a graph whose vertices are the points of , where two vertices are
adjacent Adjacent or adjacency may refer to: *Adjacent (graph theory), two vertices that are the endpoints of an edge in a graph *Adjacent (music), a conjunct step to a note which is next in the scale See also *Adjacent angles, two angles that share a c ...
if and only if they determine a line in .


Usage in statistics and econometrics

In
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, collinearity refers to a linear relationship between two
explanatory variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
s. Two variables are ''perfectly collinear'' if there is an exact linear relationship between the two, so the correlation between them is equal to 1 or −1. That is, and are perfectly collinear if there exist parameters \lambda_0 and \lambda_1 such that, for all observations , we have : X_ = \lambda_0 + \lambda_1 X_. This means that if the various observations are plotted in the plane, these points are collinear in the sense defined earlier in this article. Perfect multicollinearity refers to a situation in which explanatory variables in a multiple regression model are perfectly linearly related, according to : X_ = \lambda_0 + \lambda_1 X_ + \lambda_2 X_ + \dots + \lambda_ X_ for all observations . In practice, we rarely face perfect multicollinearity in a data set. More commonly, the issue of multicollinearity arises when there is a "strong linear relationship" among two or more independent variables, meaning that : X_ = \lambda_0 + \lambda_1 X_ + \lambda_2 X_ + \dots + \lambda_ X_ + \varepsilon_i where the variance of \varepsilon_i is relatively small. The concept of ''lateral collinearity'' expands on this traditional view, and refers to collinearity between explanatory and criteria (i.e., explained) variables.


Usage in other areas


Antenna arrays

In telecommunications, a collinear (or co-linear) antenna array is an array of
dipole antenna In radio and telecommunications a dipole antenna or doublet is the simplest and most widely used class of antenna. The dipole is any one of a class of antennas producing a radiation pattern approximating that of an elementary electric dipole w ...
s mounted in such a manner that the corresponding elements of each
antenna Antenna ( antennas or antennae) may refer to: Science and engineering * Antenna (radio), also known as an aerial, a transducer designed to transmit or receive electromagnetic (e.g., TV or radio) waves * Antennae Galaxies, the name of two collid ...
are parallel and aligned, that is they are located along a common line or axis.


Photography

The
collinearity equation The collinearity equations are a set of two equations, used in photogrammetry and computer stereo vision, to relate coordinates in a sensor plane (in two dimensions) to object coordinates (in three dimensions). The equations originate from th ...
s are a set of two equations, used in
photogrammetry Photogrammetry is the science and technology of obtaining reliable information about physical objects and the environment through the process of recording, measuring and interpreting photographic images and patterns of electromagnetic radiant ima ...
and computer stereo vision, to relate
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
in an image (
sensor A sensor is a device that produces an output signal for the purpose of sensing a physical phenomenon. In the broadest definition, a sensor is a device, module, machine, or subsystem that detects events or changes in its environment and sends ...
) plane (in two dimensions) to object coordinates (in three dimensions). In the photography setting, the equations are derived by considering the
central projection In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition, i.e., which is idempotent. The restriction to a subspace of a proje ...
of a point of the
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...
through the
optical centre In Gaussian optics, the cardinal points consist of three pairs of points located on the optical axis of a rotationally symmetric, focal, optical system. These are the '' focal points'', the principal points, and the nodal points. For ''ideal'' ...
of the camera to the image in the image (sensor) plane. The three points, object point, image point and optical centre, are always collinear. Another way to say this is that the line segments joining the object points with their image points are all concurrent at the optical centre.It's more mathematically natural to refer to these equations as ''concurrency equations'', but photogrammetry literature does not use that terminology.


See also

* Pappus's hexagon theorem * No-three-in-line problem * Incidence (geometry)#Collinearity * Coplanarity


Notes


References

* * * {{Citation , last1=Dembowski , first1=Peter , title=Finite geometries , publisher= Springer-Verlag , location=Berlin, New York , series= Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44 , mr=0233275 , year=1968 , isbn=3-540-61786-8 , url-access=registration , url=https://archive.org/details/finitegeometries0000demb Incidence geometry