HOME

TheInfoList



OR:

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 ...
, the trilinear coordinates of a point relative to a given
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
describe the relative
directed distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
s from the three sidelines of the triangle. Trilinear coordinates are an example of
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
. The ratio is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices and respectively; the ratio is the ratio of the perpendicular distances from the point to the sidelines opposite vertices and respectively; and likewise for and vertices and . In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (, , ), or equivalently in ratio form, for any positive constant . If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible for all three trilinear coordinates to be non-positive.


Notation

The ratio notation for trilinear coordinates is different from the ordered triple notation for actual directed distances. Here each of , , and has no meaning by itself; its ratio to one of the others ''does'' have meaning. Thus "comma notation" for trilinear coordinates should be avoided, because the notation , which means an ordered triple, does not allow, for example, , whereas the "colon notation" does allow .


Examples

The trilinear coordinates of the
incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
of a triangle are ; that is, the (directed) distances from the incenter to the sidelines are proportional to the actual distances denoted by , where is the inradius of . Given side lengths we have: :* :* :* :*
incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
= :*
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
= . :*
circumcenter 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 polyg ...
= . :*
orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...
= . :*
nine-point center In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circle t ...
= . :*
symmedian point In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the co ...
= . :* -excenter = :* -excenter = :* -excenter = . Note that, in general, the incenter is not the same as the
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
; the centroid has
barycentric coordinates In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
(these being proportional to actual signed areas of the triangles , where = centroid.) The midpoint of, for example, side has trilinear coordinates in actual sideline distances (0 , \tfrac , \tfrac) for triangle area , which in arbitrarily specified relative distances simplifies to . The coordinates in actual sideline distances of the foot of the altitude from to are (0, \tfrac\cos C, \tfrac\cos B), which in purely relative distances simplifies to .


Formulas


Collinearities and concurrencies

Trilinear coordinates enable many algebraic methods in triangle geometry. For example, three points :\begin P &= p:q:r \\ U &= u:v:w \\ X &= x:y:z \\ \end are
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. 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 ...
if and only if the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
: D = \begin p & q & r \\ u & v & w \\ x & y & z \end equals zero. Thus if is a variable point, the equation of a line through the points and is .William Allen Whitworth (1866
Trilinear Coordinates and Other Methods of Analytical Geometry of Two Dimensions: an elementary treatise
link from
Cornell University Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to teach an ...
Historical Math Monographs.
From this, every straight line has a linear equation homogeneous in . Every equation of the form lx+my+nz=0 in real coefficients is a real straight line of finite points unless is proportional to , the side lengths, in which case we have the locus of points at infinity. The dual of this proposition is that the lines :\begin p\alpha + q\beta + r\gamma &= 0 \\ u\alpha + v\beta + w\gamma &= 0 \\ x\alpha + y\beta + z\gamma &= 0 \end
concur In Western jurisprudence, concurrence (also contemporaneity or simultaneity) is the apparent need to prove the simultaneous occurrence of both ("guilty action") and ("guilty mind"), to constitute a crime; except in crimes of strict liability ...
in a point if and only if . Also, if the actual directed distances are used when evaluating the determinant of , then the area of triangle is , where K = \tfrac (and where is the area of triangle , as above) if triangle has the same orientation (clockwise or counterclockwise) as , and K = \tfrac otherwise.


Parallel lines

Two lines with trilinear equations lx+my+nz=0 and l'x+m'y+n'z=0 are parallel if and only if : \begin l & m & n \\ l' & m' & n' \\ a & b & c \end=0, where are the side lengths.


Angle between two lines

The
tangents In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
of the angles between two lines with trilinear equations lx+my+nz=0 and l'x+m'y+n'z=0 are given by :\pm \frac.


Perpendicular lines

Thus two lines with trilinear equations lx+my+nz=0 and l'x+m'y+n'z=0 are perpendicular if and only if :ll'+mm'+nn'-(mn'+m'n)\cos A-(nl'+n'l)\cos B-(lm'+l'm)\cos C=0.


Altitude

The equation of the
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 ...
from vertex to side is :y\cos B-z\cos C=0.


Line in terms of distances from vertices

The equation of a line with variable distances from the vertices whose opposite sides are is :apx+bqy+crz=0.


Actual-distance trilinear coordinates

The trilinears with the coordinate values being the actual perpendicular distances to the sides satisfy :aa' +bb' + cc' =2\Delta for triangle sides and area . This can be seen in the figure at the top of this article, with interior point partitioning triangle into three triangles with respective areas \tfracaa' , \tfracbb', \tfraccc'.


Distance between two points

The distance between two points with actual-distance trilinears is given by :d^2\sin ^2 C=(a_1-a_2)^2+(b_1-b_2)^2+2(a_1-a_2)(b_1-b_2)\cos C or in a more symmetric way :d^2 = \frac\left(a(b_1-b_2)(c_2-c_1)+b(c_1-c_2)(a_2-a_1)+c(a_1-a_2)(b_2-b_1)\right).


Distance from a point to a line

The distance from a point , in trilinear coordinates of actual distances, to a straight line lx+my+nz=0 is :d=\frac.


Quadratic curves

The equation of 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 specia ...
in the variable trilinear point is :rx^2+sy^2+tz^2+2uyz+2vzx+2wxy=0. It has no linear terms and no constant term. The equation of a circle of radius having center at actual-distance coordinates is :(x-a')^2\sin 2A+(y-b')^2\sin 2B+(z-c')^2\sin 2C=2r^2\sin A\sin B\sin C.


Circumconics

The equation in trilinear coordinates of any
circumconic In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.Weisstein, Eric W. "Inconic." From MathWorld- ...
of a triangle is :lyz+mzx+nxy=0. If the parameters respectively equal the side lengths (or the sines of the angles opposite them) then the equation gives 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 polyg ...
. Each distinct circumconic has a center unique to itself. The equation in trilinear coordinates of the circumconic with center is :yz(x'-y'-z')+zx(y'-z'-x')+xy(z'-x'-y')=0.


Inconics

Every conic section
inscribed {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...
in a triangle has an equation in trilinear coordinates: :l^2x^2+m^2y^2+n^2z^2 \pm 2mnyz \pm 2nlzx\pm 2lmxy =0, with exactly one or three of the unspecified signs being negative. The equation of the
incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
can be simplified to :\pm \sqrt\cos \frac\pm \sqrt\cos \frac\pm\sqrt\cos \frac=0, while the equation for, for example, the
excircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
adjacent to the side segment opposite vertex can be written as :\pm \sqrt\cos \frac\pm \sqrt\cos \frac\pm\sqrt\cos \frac=0.


Cubic curves

Many cubic curves are easily represented using trilinear coordinates. For example, the pivotal self-isoconjugate cubic , as the locus of a point such that the -isoconjugate of is on the line is given by the determinant equation : \beginx&y&z\\ qryz&rpzx&pqxy\\u&v&w\end = 0. Among named cubics are the following: :
Thomson cubic In geometry, the Thomson cubic of a triangle is the locus of centers of circumconics whose normals at the vertices are concurrent Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to: Law * Concur ...
: ''Z(X(2),X(1))'', where ''X(2) = ''
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
, ''X(1) = ''
incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
: Feuerbach cubic: ''Z(X(5),X(1))'', where ''X(5) = ''
Feuerbach point In the geometry of triangles, the incircle and nine-point circle of a triangle are internally tangent to each other at the Feuerbach point of the triangle. The Feuerbach point is a triangle center, meaning that its definition does not depend on th ...
:
Darboux cubic In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an equ ...
: ''Z(X(20),X(1))'', where ''X(20) = ''
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 ...
:
Neuberg cubic In Euclidean geometry, the Neuberg cubic is a special cubic plane curve associated with a reference triangle with several remarkable properties. It is named after Joseph Jean Baptiste Neuberg (30 October 1840 – 22 March 1926), a Luxembourger mat ...
: ''Z(X(30),X(1))'', where ''X(30) = ''
Euler infinity point Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
.


Conversions


Between trilinear coordinates and distances from sidelines

For any choice of trilinear coordinates to locate a point, the actual distances of the point from the sidelines are given by where can be determined by the formula k = \tfrac in which are the respective sidelengths , and is the area of .


Between barycentric and trilinear coordinates

A point with trilinear coordinates has
barycentric coordinates In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
where are the sidelengths of the triangle. Conversely, a point with barycentrics has trilinear coordinates \tfrac : \tfrac : \tfrac.


Between Cartesian and trilinear coordinates

Given a reference triangle , express the position of the vertex in terms of an ordered pair of
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
and represent this algebraically as a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
using vertex as the origin. Similarly define the position vector of vertex as Then any point associated with the reference triangle can be defined in a Cartesian system as a vector \vec P = k_1\vec A + k_2\vec B. If this point has trilinear coordinates then the conversion formula from the coefficients and in the Cartesian representation to the trilinear coordinates is, for side lengths opposite vertices , : x:y:z = \frac : \frac : \frac, and the conversion formula from the trilinear coordinates to the coefficients in the Cartesian representation is : k_1 = \frac, \quad k_2 = \frac. More generally, if an arbitrary origin is chosen where the Cartesian coordinates of the vertices are known and represented by the vectors and if the point has trilinear coordinates , then the Cartesian coordinates of are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates as the weights. Hence the conversion formula from the trilinear coordinates to the vector of Cartesian coordinates of the point is given by : \vec=\frac\vec+\frac\vec+\frac\vec, where the side lengths are :\begin & , \vec C - \vec B, = a, \\ & , \vec A - \vec C, = b, \\ & , \vec B - \vec A, = c. \end


See also

* Morley's trisector theorem#Morley's triangles, giving examples of numerous points expressed in trilinear coordinates *
Ternary plot A ternary plot, ternary graph, triangle plot, simplex plot, Gibbs triangle or de Finetti diagram is a barycentric plot on three variables which sum to a constant. It graphically depicts the ratios of the three variables as positions in an equi ...
*
Viviani's theorem Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from ''any'' interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. It is a theorem commonly employed in various ma ...


References


External links

*
Encyclopedia of Triangle Centers - ETC
by Clark Kimberling; has trilinear coordinates (and barycentric) for more than 7000 triangle centers {{Authority control Linear algebra Affine geometry Triangle geometry Coordinate systems