BarycentricCoordinates

1. Preliminaries

In the subsequent discussion points A,B, ... of the plane are identified with two dimensional vectors (a₁,a₂), (b₁,b₂) ... .

[1] Three points of the plane {A,B,C} are collinear <==>
There are three numbers (x,y,z) not all zero, such that xA+yB+cZ = 0 and x+y+z=0.
(<= part) If x+y+z=0 and xA+yB+zC=0 => (-y-z)A+yB+zC=0 => y(B-A)+z(C-A)=0 i.e. {A,B,C} collinear.
(=> part) Reverse the above implications.

[2] Let the points of the plane {A,B,C} be non-collinear.
Then for every other point D of the plane there are unique numbers (x,y,z) with x+y+z=1 and D=xA+yB+zC.
In fact, extend the coordinates the two-dimensional vectors (x,y) to three dimensional (x,y,1). Denote the corresponding space-points by {A',B',C',D'}.
(i) Apply [1] to see that {A',B',C'} are independent.
(ii) Express D' in the basis {A',B',C'} to find the uniquely defined (x,y,z) as required.
[3] Given three numbers x, y and z, point E is defined through the vector equation:
OE = x*OA + y*OB + z*OC.
It is easily seen that E depends on the position of the origin O and the three numbers (x,y,z).

Taking s = x+y+z and x' = x/s, y' = y/s, z' = z/s one defines D (collinear with {O,E}) through the same equation:
OD = x'*OA + y'*OB + z'*OC (now (x',y',z') satisfying x'+y'+z'=1).

[4] For non-collinear {A,B,C} point D, defined by the above equation, is independent of the position of O and depends only on the three numbers (x',y',z') with x'+y'+z'=1.
In fact, using another point O' for origin the same equation and the analogous numbers (x'',y'',z'') to express D: O'D = x''*O'A + y''*O'B + z''*O'C and substracting we get:
OD-O'D = (x'*OA + y'*OB + z'*OC)-(x''*O'A + y''*O'B + z''*O'C) =
= (x'-x'')A + (y'-y'')B + (z'-z'')C - (x'O-x''O' + y'O-y''O' + z'O-z''O') =>
OO' = O'-O = (x'-x'')A + (y'-y'')B + (z'-z'')C - (O-O') =>
(x'-x'')A + (y'-y'')B + (z'-z'')C = 0.
By [1] this implies (x'-x'')=0, (y'-y'')=0, (z'-z'')=0, as desired.

[5] The numbers (x',y',z') with x'+y'+z'=1 expressing D in the above equation are equal to the quotients of signed areas:
x' = Area(DBC)/Area(ABC), y' = Area(DCA)/Area(ABC), z' = Area(DAB)/Area(ABC).
Using the independence of (x',y',z') from the position of O we select O=D and use x'=1-y'-z' =>
(1-y'-z')DA+y'DB+z'DC=0 => DA + y'(DB-DA)+z'(DC-DA)=0 => AD = y'(AB) + z'(AC).
Multiplying externally by AB => (AD)x(AB) = z'(AC x AB) or equivalently
z'(AB x AC) = (AB x AD).
Analogous equations result also for y' and x'. Taking the positive orientation in the direction ABxAD we have the result stated.

2. Barycentric Coordinates - definition

Let {A,B,C} be three non-collinear points of the plane. For every other point D of the plane there are unique numbers (x,y,z) satisfying x+y+z=1 such that D=xA+yB+zC. Numbers (x,y,z) are given by the quotients of signed areas:
x = Area(DBC)/Area(ABC), y = Area(DCA)/Area(ABC), z = Area(DAB)/Area(ABC).

(x,y,z) are called absolute barycentric homogeneous coordinates of point D. More general, one considers multiples (c*x, c*y, c*z) as general barycentric homogeneous coordinates of point D.
The signs of the numbers result from the orientations of the corresponding triangles e.g. if {D,A} are on the same side of BC then x is positive. Otherwise it is negative and for D on BC it is zero. In fact x=0 characterizes line BC. Analogous remarks hold also for y and z.

3. Traces and ratios

Varying x alone, moves D along a fixed line through A. For x = 0, we get D_A on the line BC, identified with the point D_A = yB+zC. D_A is called the trace of D on side BC of the triangle ABC. Analogously are defined the traces D_B and D_C.
The oriented ratio D_AB/D_AC, calculated using y+z=1, gives the value -(z/y).

4. Projective Geometry aspect

The barycenter of the triangle, i.e. the intersection-point G of its medians, defines triangles GBC, GCA, GAB with equal areas, thus it has homogeneous barycentric coordinates (1,1,1). Thus, barycentric coordinates coincide with projective coordinates with respect to the projective basis (A,B,C,G) (see ProjectiveBasis.html ).

It follows that the homogeneous barycentric coordinates use the vertices (A,B,C) of the triangle and its centroid G to define a projective map of the standard two-dimensional real projective plane P² to the projectified (extended affine) plane of the triangle. Denote this (projective) plane by Q. Such a map is completely determined by 4 projectively independent points and their images. In our case the four points of P² are defined through the standard basis A*=[e₁], B*=[e₂], C*=[e₃] and M*=[e₁+e₂+e₃]. Their corresponding images being the points A, B, C and M. Denote this projective map by F. Point X of P², represented by the homogeneous coordinates [x,y,z], maps via F to point xA+yB+zC = D.

Every line ax+by+cz=0 of P² maps under F on a line of Q expressed through the same equation. In particular the projective line of P² defined by the equation x+y+z=0, maps to the corresponding line at infinity of Q expressed also by the equation x+y+z=0.

Consider the plane S of three dimensional space defined by the equation: x+y+z =1, and passing through the points e₁, e₂, e₃. The projective map F, considered above, is the "projective extension" of the affine map F* of S on Q, mapping e₁ to A, e₂ to B and e₃ to C.
In S, points e₁, e₂, e₃ are vertices of an equilateral triangle and g=(e₁+e₂+e₃)/3 is the center (of gravity) of this equilateral, mapping under F onto the barycenter of triangle ABC.

Thus barycentric coordinates transfer the apparatus of projective geometry to the geometry of the triangle ABC, enabling the simple formulation/investigation of coincidence relations.

5. References

Berger, M Geometry I, II Paris, Springer Verlag, 1987, vol. I, p. 123, 140.
Bradley, J. Christopher Challenges in Geometry Oxford, Oxford University Press 2005, p.123.
Pedoe, D. A course of Geometry, for colleges and universities London, Cambridge University Press, 1970, p.113
Yiu, P. GeometryNotes020402 http://www.math.fau.edu/yiu/GeometryNotes020402.pdf
Yiu, P. Euclidean Geometry Notes http://www.math.fau.edu/yiu/EuclideanGeometryNotes.pdf

1. Preliminaries

2. Barycentric Coordinates - definition

3. Traces and ratios

4. Projective Geometry aspect

5. References

See Also