"Colinear" redirects here. For the use in genetics, see
synteny. For the use in coalgebra theory, see
colinear map. For colinearity in statistics, see
multicollinearity.
In
geometry, collinearity of a set of
points is the property of their lying on a single
line.[1] A set of points with this property is said to be collinear (sometimes spelled as colinear[2]). 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 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, 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:
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.[3]: p.199
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.[4]: p.120, #78
Menelaus' theorem states that three points on the sides (some
extended) of a triangle opposite vertices respectively are collinear if and only if the following products of segment lengths are equal:[3]: p. 147
The incenter, the centroid, and the Spieker circle's center are collinear.
In a convex
quadrilateralABCD whose opposite sides intersect at E and F, the
midpoints of AC, BD, EF are collinear and the line through them is called the
Newton line. If the quadrilateral is a
tangential quadrilateral, then its incenter also lies on this line.[6]
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,
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, 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, 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 n-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
Equivalently, for every subset of X, Y, Z, if the
matrix
is of
rank 2 or less, the points are collinear. In particular, for three points in the plane (n = 2), 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 A, B, C, the following determinant of a
Cayley–Menger determinant is zero (with d(AB) meaning the distance between A and B, etc.):
This determinant is, by
Heron's formula, equal to −16 times the square of the area of a triangle with side lengths d(AB), d(BC), d(AC); so checking if this determinant equals zero is equivalent to checking whether the triangle with vertices A, B, C 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 A, B, C with d(AC) greater than or equal to each of d(AB) and d(BC), the
triangle inequalityd(AC) ≤ d(AB) + d(BC) holds with equality.
Number theory
Two numbers m and n 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 (0, 0), (m, 0), (m, n), (0, n), at least one interior point is collinear with (0, 0) and (m, n).
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 geometryP, where two points determine at most one line, a collinearity graph of P is a
graph whose vertices are the points of P, where two vertices are
adjacent if and only if they determine a line in P.
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, X1 and X2 are perfectly collinear if there exist parameters and such that, for all observations i, we have
This means that if the various observations (X1i, X2i) are plotted in the (X1, X2) plane, these points are collinear in the sense defined earlier in this article.
Perfect multicollinearity refers to a situation in which k (k ≥ 2) explanatory variables in a
multiple regression model are perfectly linearly related, according to
for all observations i. 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.[10]
The
collinearity equations are a set of two equations, used in
photogrammetry and
computer stereo vision, to relate
coordinates 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 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.[11]