Trilinear coordinates

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Trilinear coordinates.svg

In geometry, the trilinear coordinates x:y:z of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. They are often called simply "trilinears". The ratio x:y is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices A and B respectively; the ratio y:z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C respectively; and likewise for z:x and vertices C and A.

In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (a' , b' , c' ), or equivalently in ratio form, ka' :kb' :kc' for any positive constant k. 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 trilinears to be non-positive.

Notation[edit]

The ratio notation x:y:z for trilinears is different from the ordered triple notation (a' , b' , c' ) for actual directed distances. ("Comma notation" for trilinears should be avoided, because the notation (x, y, z), which means an ordered triple, does not allow, for example, (x, y, z) = (2x, 2y, 2z), whereas the "colon notation" does allow x : y : z = 2x : 2y : 2z.)

Examples[edit]

The trilinears of the incenter of a triangle ABC are 1 : 1 : 1; that is, the (directed) distances from the incenter to the sidelines BC, CA, AB are proportional to the actual distances denoted by (r, r, r), where r is the inradius of triangle ABC. Given side lengths a, b, c we have:

  • A = 1 : 0 : 0
  • B = 0 : 1 : 0
  • C = 0 : 0 : 1
  • incenter = 1 : 1 : 1
  • centroid = bc : ca : ab = 1/a : 1/b : 1/c = csc A : csc B : csc C.
  • circumcenter = cos A : cos B : cos C.
  • orthocenter = sec A : sec B : sec C.
  • nine-point center = cos(BC) : cos(CA) : cos(AB).
  • symmedian point = a : b : c = sin A : sin B : sin C.
  • A-excenter = −1 : 1 : 1
  • B-excenter = 1 : −1 : 1
  • C-excenter = 1 : 1 : −1.

Note that, in general, the incenter is not the same as the centroid; the centroid has barycentric coordinates 1 : 1 : 1 (these being proportional to actual signed areas of the triangles BGC, CGA, AGB, where G = centroid.)

The midpoint of, for example, side BC has trilinear coordinates in actual sideline distances (0 , \frac{\Delta}{b} , \frac{\Delta}{c}), which in arbitrarily specified relative distances simplifies to 0:ca:ab. The coordinates in actual sideline distances of the foot of the altitude from A to BC are (0, \frac{2\Delta}{a}\cos C, \frac{2\Delta}{a}\cos B) for triangle area \Delta, which in purely relative distances simplifies to 0:\cos C:\cos B.[1]:p. 96

Formulas[edit]

Midpoints[edit]

The midpoint between two points with trilinear coordinates x : y : z and x' : y' : z' is \frac{x+x'}{2}:\frac{y+y'}{2}:\frac{z+z'}{2}. [1]:p.97, #5

Collinearities and concurrencies[edit]

Trilinears enable many algebraic methods in triangle geometry. For example, three points

P = p : q : r
U = u : v : w
X = x : y : z

are collinear if and only if the determinant

 D = \begin{vmatrix}p&q&r\\
u&v&w\\x&y&z\end{vmatrix}

equals zero. Thus if x:y:z is a variable point, the equation of a line through the points P and U is D = 0.[1]:p. 23 From this, every straight line has a linear equation homogeneous in x, y, z. Every equation of the form lx+my+nz = 0 in real coefficients is a real straight line of finite points unless l : m: n is proportional to a : b : c, the side lengths, in which case we have the locus of points at infinity.[1]:p. 40

The dual of this proposition is that the lines

pα + qβ + rγ = 0
uα + vβ + wγ = 0,
xα + yβ + zγ = 0

concur in a point (α, β, γ) if and only if D = 0.[1]:p. 28

Also, if the actual directed distances are used when evaluating the determinant of D, then (area of (PUX)) = KD, where K = abc/8∆2 if triangle PUX has the same orientation as triangle ABC, and K = - abc/8∆2 otherwise.

Parallel lines[edit]

Two lines with trilinear equations lx+my+nz=0 and l'x+m'y+n'z=0 are parallel if and only if[1]:p. 98,#xi

 \begin{vmatrix}l&m&n\\
l'&m'&n'\\a&b&c\end{vmatrix}=0,

where a, b, c are the side lengths.

Perpendicular lines[edit]

Two lines with trilinear equations lx+my+nz=0 and l'x+m'y+n'z=0 are perpendicular if and only if[1]:p. 50

ll'+mm'+nn'-(mn'+m'n)\cos A-(nl'+n'l)\cos B-(lm'+l'm)\cos C=0.

Altitude[edit]

The equation of the altitude from vertex A to side BC is[1]:p.98,#x

y\cos B-z\cos C=0.

Line with given distances from vertices[edit]

The equation of a line with distances p, q, r from the vertices A, B, C whose opposite sides are a, b, c is[1]:p. 97,#viii

apx+bqy+crz=0.

Actual distance trilinears[edit]

The trilinears with the coordinate values a', b', c' being the actual perpendicular distances to the sides satisfy[1]:p. 11

aa' +bb' + cc' =2\Delta

for triangle sides a, b, c and area \Delta. This can be seen in the figure at the top of this article, with interior point P partitioning triangle ABC into three triangles PBC, PCA, and PAB with respective areas (1/2)aa' , (1/2)bb' , and (1/2)cc' .

Distance between two points[edit]

The distance d between two points with actual distance trilinears a' i : b' i : c' i is given by[1]:p. 46

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.

Distance from a point to a line[edit]

The distance d from a point a' 0 : b' 0 : c' 0, in trilinear coordinates of actual distances, to a straight line lx + my + nz = 0 is[1]:p. 48

d=\frac{la'_0+mb'_0+nc'_0}{\sqrt{l^2+m^2+n^2-2mn\cos A -2nl\cos B -2lm\cos C}}.

Quadratic curves[edit]

The equation of a conic section in the variable trilinear point x : y : z is[1]:p.118

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 r having center at actual-distance coordinates (a', b', c' ) is[1]:p.287

(x-a')^2\sin 2A+(y-b')^2\sin 2B+(z-c')^2\sin 2C=2r^2\sin A\sin B\sin C.

Circumconics[edit]

The equation in trilinears x, y, z of any circumconic of a triangle is[1]:p. 192

lyz+mzx+nxy=0.

If the parameters l, m, n respectively equal the side lengths a, b, c (or the sines of the angles opposite them) then the equation gives the circumcircle.[1]:p. 199

Each distinct circumconic has a center unique to itself. The equation in trilinears of the circumconic with center x' : y' : z' is[1]:p. 203

yz(x'-y'-z')+zx(y'-z'-x')+xy(z'-x'-y')=0.

Inconics[edit]

Every conic section inscribed in a triangle has equation in trilinears[1]:p. 208

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 can be simplified to[1]:p. 210, p.214

\pm \sqrt{x}\cos \frac{A}{2}\pm \sqrt{y}\cos \frac{B}{2}\pm\sqrt{z}\cos \frac{C}{2}=0,

while the equation for, for example, the excircle adjacent to the side segment opposite vertex A can be written as[1]:p. 215

\pm \sqrt{-x}\cos \frac{A}{2}\pm \sqrt{y}\cos \frac{B}{2}\pm\sqrt{z}\cos \frac{C}{2}=0.

Cubic curves[edit]

Many cubic curves are easily represented using trilinears. For example, the pivotal self-isoconjugate cubic Z(U,P), as the locus of a point X such that the P-isoconjugate of X is on the line UX is given by the determinant equation

 \begin{vmatrix}x&y&z\\
qryz&rpzx&pqxy\\u&v&w\end{vmatrix} = 0.

Among named cubics Z(U,P) are the following:

Thomson cubic: Z(X(2),X(1)), where X(2) = centroid, X(1) = incenter
Feuerbach cubic: Z(X(5),X(1)), where X(5) = Feuerbach point
Darboux cubic: Z(X(20),X(1)), where X(20) = De Longchamps point
Neuberg cubic: Z(X(30),X(1)), where X(30) = Euler infinity point.

Conversions[edit]

Between trilinear coordinates and distances from sidelines[edit]

For any choice of trilinear coordinates x:y:z to locate a point, the actual distances of the point from the sidelines are given by a' = kx, b' = ky, c' = kz where k can be determined by the formula k = \frac{2\Delta}{ax + by + cz} in which a, b, c are the respective sidelengths BC, CA, AB, and ∆ is the area of ABC.

Between barycentric and trilinear coordinates[edit]

A point with trilinears x : y : z has barycentric coordinates ax : by : cz where a, b, c are the sidelengths of the triangle. Conversely, a point with barycentrics α : β : γ has trilinears α/a : β/b : γ/c.

Between Cartesian and trilinear coordinates[edit]

Given a reference triangle ABC, express the position of the vertex B in terms of an ordered pair of Cartesian coordinates and represent this algebraically as a vector B, using vertex C as the origin. Similarly define the position vector of vertex A as A. Then any point P associated with the reference triangle ABC can be defined in a Cartesian system as a vector P = k1A + k2B. If this point P has trilinear coordinates x : y : z then the conversion formula from the coefficients k1 and k2 in the Cartesian representation to the trilinear coordinates is, for side lengths a, b, c opposite vertices A, B, C,

x:y:z = \frac{k_1}{a} : \frac{k_2}{b} : \frac{1 - k_1 - k_2}{c},

and the conversion formula from the trilinear coordinates to the coefficients in the Cartesian representation is

k_1 = \frac{ax}{ax + by + cz}, \quad k_2 = \frac{by}{ax + by + cz}.

More generally, if an arbitrary origin is chosen where the Cartesian coordinates of the vertices are known and represented by the vectors A, B and C and if the point P has trilinear coordinates x : y : z, then the Cartesian coordinates of P are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates ax, by and cz as the weights. Hence the conversion formula from the trilinear coordinates x, y, z to the vector of Cartesian coordinates P of the point is given by

\underline{P}=\frac{ax}{ax+by+cz}\underline{A}+\frac{by}{ax+by+cz}\underline{B}+\frac{cz}{ax+by+cz}\underline{C},

where the side lengths are |CB| = a, |AC| = b and |BA| = c.

See also[edit]

References[edit]

  1. ^ a b c d e f g h i j k l m n o p q r s t Whitworth, William Allen. Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions, Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866). http://www.forgottenbooks.com/search?q=Trilinear+coordinates&t=books

External links[edit]