In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
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
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pr ...
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, 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 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
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every pol ...
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
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
, 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
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
respectively are collinear if and only if the following products of segment lengths are equal:
[
::
* The incenter, the centroid, and the Spieker circle's center are collinear.
*The circumcenter, the Brocard midpoint, and the Lemoine point 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 sides. 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, 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 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
In mathematics, a conic section, quadratic curve or conic 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 ...
(i.e., ellipse, parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descri ...
or hyperbola) and joined by line segments in any order to form a hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A ''regular hexagon'' h ...
, 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, for any three circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
s 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, the center, the two foci, 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 of a conic solid 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 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
:
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'' (franchi ...
:
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'' (franchi ...
:
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 is zero (with meaning the distance between and , etc.):
::
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 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 , 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, collinearity refers to a linear relationship between two explanatory variables. 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 and such that, for all observations , we have
:
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
:
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
:
where the variance of 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 telecommunication
Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than tha ...
s, a collinear (or co-linear) antenna array is an array of dipole antennas mounted in such a manner that the corresponding elements of each antenna 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 the ...
s are a set of two equations, used in photogrammetry 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 si ...
in an image ( sensor) plane (in two dimensions) to object coordinates (in three dimensions). In the photography setting, the equations are derived by considering the central projection of a point of the object through the optical centre of the camera
A camera is an optical instrument that can capture an image. Most cameras can capture 2D images, with some more advanced models being able to capture 3D images. At a basic level, most cameras consist of sealed boxes (the camera body), with a ...
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