Trilinear Coordinates

Given a reference triangle DeltaABC, the trilinear coordinates of a point P with respect to DeltaABC are an ordered triple of numbers, each of which is proportional to the directed distance from P to one of the side lines. Trilinear coordinates are denoted alpha:beta:gamma or (alpha,beta,gamma) and also are known as homogeneous coordinates or "trilinears." Trilinear coordinates were introduced by Plücker in 1835. Since it is only the ratio of distances that is significant, the triplet of trilinear coordinates obtained by multiplying a given triplet by any nonzero constant describes the same point, so


For simplicity, the three polygon vertices A, B, and C of a triangle are commonly written as 1:0:0, 0:1:0, and 0:0:1, respectively.


Trilinear coordinates can be normalized so that they give the actual directed distances from P to each of the sides. To perform the normalization, let the point P in the above diagram have trilinear coordinates alpha:beta:gamma and lie at distances a^', b^', and c^' from the sides BC, AC, and AB, respectively. Then the distances a^'=kalpha, b^'=kbeta, and c^'=kgamma can be found by writing Delta_a for the area of DeltaBPC, and similarly for Delta_b and Delta_c. We then have




where Delta is the area of DeltaABC and a, b, and c are the lengths of its sides (Kimberling 1998, pp. 26-27). To obtain trilinear coordinates giving the actual distances, take k=1, so we have the coordinates


These normalized trilinear coordinates are known as exact trilinear coordinates.

The trilinear coordinates of the line




where d_i is the point-line distance from polygon vertex i to the line.

The homogeneous barycentric coordinates corresponding to trilinear coordinates alpha:beta:gamma are (aalpha,bbeta,cgamma), and the trilinear coordinates corresponding to homogeneous barycentric coordinates (t_1,t_2,t_3) are t_1/a:t_2/b:t_3/c.

Important points alpha:beta:gamma of a triangle are called triangle centers, and the vector functions describing the location of the points in terms of side length, angles, or both, are called triangle center functions f(a,b,c). Since by symmetry, triangle center functions are of the form


it is common to call the scalar function f(a,b,c) "the" triangle center function. Note also that side lengths and angles are interconvertible through the law of cosines, so a triangle center function may be given in terms of side lengths, angles, or both. Trilinear coordinates for some common triangle centers are summarized in the following table, where A, B, and C are the angles at the corresponding vertices and a, b, and c are the opposite side lengths. Here, the normalizations have been chosen to give a simple form.


In trilinear coordinates, the coordinates of the vertices are 1:0:0 (A), 0:1:0 (B), and 0:0:1 (C). Extensions along the sidelines by a distance d have trilinears as illustrated above.


Trilinear coordinates of points fractional distances k_a, k_b, and k_c along the sidelines are given in the above figure, where k_i^'=1-k_i.

A point located a fraction k of the distance along the sideline AC from A to C has trilinear coordinates


To determine the conversion of trilinear to Cartesian coordinates, orient the triangle with the BC axis parallel to the x-axis and with its incenter at the origin, as illustrated above. Then




is the inradius, Delta is the triangle area, and


(Kimberling 1998, pp. 31-33).

More generally, to convert trilinear coordinates to a vector position for a given triangle specified by the x- and y-coordinates of its axes, pick two unit vectors along the sides. For instance, pick

a^^=[a_1; a_2]
c^^=[c_1; c_2],

where these are the unit vectors BC and AB. Assume the triangle has been labeled such that A=x_1 is the upper rightmost polygon vertex and C=x_2. Then the vectors obtained by traveling l_a and l_c along the sides and then inward perpendicular to them must meet

 [x_1; y_1]+l_c[c_1; c_2]-kgamma[c_2; -c_1]=[x_2; y_2]+l_a[a_1; a_2]-kalpha[a_2; -a_1].

Solving the two equations




But a^^ and c^^ are unit vectors, so


And the vector coordinates of the point alpha:beta:gamma are then

 x=x_1+l_c[c_1; c_2]-kgamma[c_2; -c_1].

See also

Areal Coordinates, Barycentric Coordinates, Exact Trilinear Coordinates, Major Triangle Center, Orthocentric Coordinates, Power Curve, Quadriplanar Coordinates, Reference Triangle, Regular Triangle Center, Triangle, Triangle Center, Triangle Center Function, Trilinear Line, Trilinear Polar, Trilinear Vertex Matrix, Tripolar Coordinates

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Boyer, C. B. History of Analytic Geometry. New York: Yeshiva University, 1956.Casey, J. "The General Equation--Trilinear Co-Ordinates." Ch. 10 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 333-348, 1893.Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 67-71, 1959.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.Coxeter, H. S. M. "Some Applications of Trilinear Coordinates." Linear Algebra Appl. 226-228, 375-388, 1995.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Wong, M. K. F. Int. J. Math. Educ. Sci. Tech. 27, 293-296, 1996.Wong, M. K. F. Int. J. Math. Educ. Sci. Tech. 29, 143-145, 1998.

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Trilinear Coordinates

Cite this as:

Weisstein, Eric W. "Trilinear Coordinates." From MathWorld--A Wolfram Web Resource.

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