"Colinear" redirects here. For the use in genetics, seesynteny. For the use in coalgebra theory, seecolinear map. For colinearity in statistics, seemulticollinearity.
Ingeometry,collinearity of a set ofpoints is the property of their lying on a singleline.[1] A set of points with this property is said to becollinear (sometimes spelled ascolinear[2]). In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".
In any geometry, the set of points on a line are said to becollinear. InEuclidean geometry this relation is intuitively visualized by points lying in a row on a "straight line". However, in most geometries (including Euclidean) aline is typically aprimitive (undefined) object type, so such visualizations will not necessarily be appropriate. Amodel 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, inspherical 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 beingin a row.
A mapping of a geometry to itself which sends lines to lines is called acollineation; it preserves the collinearity property. Thelinear maps (or linear functions) ofvector spaces, viewed as geometric maps, map lines to lines; that is, they map collinear point sets to collinear point sets and so, are collineations. Inprojective geometry these linear mappings are calledhomographies and are just one type of collineation.
Any vertex, the tangency of the opposite side with the incircle, and theGergonne point are collinear.
From any point on thecircumcircle of a triangle, the nearest points on each of the three extended sides of the triangle are collinear in theSimson line of the point on the circumcircle.
The lines connecting the feet of thealtitudes intersect the opposite sides at collinear points.[3]: p.199
A triangle'sincenter, the midpoint of analtitude, and the point of contact of the corresponding side with theexcircle relative to that side are collinear.[4]: p.120, #78
Menelaus' theorem states that three points on the sides (someextended) 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 convexquadrilateralABCD whose opposite sides intersect atE andF, themidpoints ofAC,BD,EF are collinear and the line through them is called theNewton line. If the quadrilateral is atangential quadrilateral, then its incenter also lies on this line.[6]
Pascal's theorem (also known as the Hexagrammum Mysticum Theorem) states that if an arbitrary six points are chosen on aconic section (i.e.,ellipse,parabola orhyperbola) and joined by line segments in any order to form ahexagon, 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: theBraikenridge–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 inPappus's hexagon theorem.
ByMonge's theorem, for any threecircles 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 anellipse, the center, the twofoci, and the twovertices with the smallestradius of curvature are collinear, and the center and the two vertices with the greatest radius of curvature are collinear.
In ahyperbola, the center, the two foci, and the two vertices are collinear.
Thecenter of mass of aconic 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.
The centroid of a tetrahedron is the midpoint between itsMonge point andcircumcenter. These points define theEuler line of the tetrahedron that is analogous to theEuler line of a triangle. The center of thetetrahedron's twelve-point sphere also lies on the Euler line.
Incoordinate geometry, inn-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 ofrank 1 or less. For example, given three points
Equivalently, for every subset ofX, Y, Z, if thematrix
is ofrank 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 itsdeterminant is zero; since that 3 × 3 determinant is plus or minus twice thearea 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 calledstraight, meaning all the points are collinear, if and only if, for every three of those pointsA, B, C, the following determinant of aCayley–Menger determinant is zero (withd(AB) meaning the distance betweenA andB, etc.):
This determinant is, byHeron's formula, equal to −16 times the square of the area of a triangle with side lengthsd(AB),d(BC),d(AC); so checking if this determinant equals zero is equivalent to checking whether the triangle with verticesA, 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 pointsA, B, C withd(AC) greater than or equal to each ofd(AB) andd(BC), thetriangle inequalityd(AC) ≤d(AB) +d(BC) holds with equality.
Two numbersm andn are notcoprime—that is, they share a common factor other than 1—if and only if for a rectangle plotted on asquare 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).
In variousplane geometries the notion of interchanging the roles of "points" and "lines" while preserving the relationship between them is calledplane 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 calledconcurrency, and the lines are said to beconcurrent lines. Thus, concurrency is the plane dual notion to collinearity.
Given apartial geometryP, where two points determine at most one line, acollinearity graph ofP is agraph whose vertices are the points ofP, where two vertices areadjacent if and only if they determine a line inP.
Instatistics,collinearity refers to a linear relationship between twoexplanatory variables. Two variables areperfectly 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 andX2 are perfectly collinear if there exist parameters and such that, for all observationsi, 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.
Perfectmulticollinearity refers to a situation in whichk (k ≥ 2) explanatory variables in amultiple regression model are perfectly linearly related, according to
for all observationsi. 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 oflateral collinearity expands on this traditional view, and refers to collinearity between explanatory and criteria (i.e., explained) variables.[10]
Thecollinearity equations are a set of two equations, used inphotogrammetry andcomputer stereo vision, to relatecoordinates in an image (sensor) plane (in two dimensions) to object coordinates (in three dimensions). In the photography setting, the equations are derived by considering thecentral projection of a point of theobject through theoptical centre of thecamera 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]
