LineCoordinates

1. Line coordinates

The lines define a triangle t = A₁A₂A₃, such that A_i is opposite to the line f_i(x,y)=0.
Further it is assumed that the lines are in normal form i.e. their coefficients defining their normal
build a unit-length vector ( a_i²+b_i²=1 ).
Every other line represented by an equation f(x,y) = ax+by+c =0 can be written as a combination:

Written in short vector notation a^t = k^t M.
The numbers (k₁,k₂,k₃) are called the line coordinates of the line with respect to the triangle
defined by the three lines {f_i(x,y)=0}. Thus a line defines three line coordinates, but the inverse is
not true. Not every triple (k₁,k₂,k₃) defines a line in this way. See section 4 below.
Besides the line coordinates are unique up to a non-zero multiplicative constant k, since f(x,y)=0 is
equivalent to kf(x,y)=0. See ThreeLines.html for the geometric meaning of (k_i).
Remark Note that matrix M expresses also the relation between cartesian (x,y,1)^t and trilinear
coordinates (see Trilinears.html ) (X,Y,Z)^t:

2. Matrix splitting

There is useful matrix splitting for the matrix N resulting from the above one by multiplying with the transpose:
(1)                                                                     N = MM^t.
The (i,j)-entry of  the symmetric matrix N is an interesting inner product plus a correction term:
(2)                                                                      n_ij =  a_ia_j+b_ib_j+c_ic_j = <n_i, n_j> + c_ic_j.
With  n_i I denote the two-dimensional vector (a_i, b_i)^t giving the normal of the i-th line f_i(x,y)=0.
By selecting the origin of coordinates inside the triangle and considering the angles of the unit-vectors it is readily
seen that their inner products for i different from j satisfy:
(3)                                                                     <n_i,n_j> = -cos(A_k),
where k is the other index of the three than (i,j). Thus the matrix N has the form:

3. Angle of two lines

Using the above splitting one can easily find the expression of the angle of two lines in line-coordinates.
In fact, the inner product aa'+bb'+cc' of the coefficients of the two lines:
f(x,y) = ax+by+c, f'(x,y) = a'x+b'y+c',
is easily expressible through they corresponding coefficients (k_i) and (k'_i) giving (Salmon p. 59):

The determinant of the above matrix Q is easily found to be zero:
|Q| = 1-2cos(A₁)cos(A₂)cos(A₃)-cos²(A₁)-cos²(A₂)-cos²(A₃)=0.
The two-dimensional vectors (a,b) express the coordinates of the normal n of the corresponding line.
Thus the inner product (aa'+bb') gives the cosine of the angle between the lines, if the normals have unit length.
The quadratic form q(k,k') = k^t Q k' defined by the previous singular matrix Q dominates all formulas involving metric
relations in terms of trilinear coordinates.
See for example [Carnoy, pp. 76-89] who uses the notation {k₁,k₂,k₃} for q(k,k).

4. Exceptional coefficients

Not every triple (k₁,k₂,k₃) defines a line. The multiples of the triple (a,b,c) of the side-lengths of the triangle
must be excluded since they define always a constant:
(4) af₁(x,y) + bf₂(x,y) + cf₃(x,y) = 2D.
Here D is the area of the triangle of reference and the equality is valid because for every point (x,y), the value
f_i(x,y) gives the signed distance from the side-line f_i(x,y)=0 (because of the normalization of the equation).
Thus, equation (4) is satisfied for every point of the plane.

5. Homogenization of line equation

Equation (4) allows to write every line equation in line-coordinates in a homogeneous way with respect to
the line coordinates (k₁,k₂,k₃). For example the equation:

Thus in line-coordinates it can be assumed from a line that it has no constant term, like the "c" in the
cartesian equation ax+by+c=0.

6. The line at infinity

Writing the usual line equation in cartesian coordinates ax'+by'+c=0, and introducing the additional variable z
through x'=x/z, y'=y/z one obtains the homogeneous line equation ax+by+cz=0. All the ordinary points
are characterized by the equation z=1. The equation z=0 describes the line at infinity. This is a set of additional
points that complements the euclidean plane and turns it to a projective plane.
Analogous is the interpretation of the equation (4). It characterizes all ordinary points of the plane. Equation:
(5) af₁(x,y)+bf₂(x,y)+cf₃(x,y) = 0,
characterizes though the points of the line at infinity.

7. Interpretation of constants

Within the extension of the plane through the homogeneous coordinates the impossible equation c=0 in the
euclidean plane can be interpreted in the projective as the simplification of  cz=0 i.e. the equation of the line
at infinity. The family of parallel lines ax+by+c=0 resulting by varying c can be interpreted as the pencil of
lines combining  the ordinary line  ax+by=0 with the line at infinity:  (ax+by) + c(z) = 0. All the lines of the
pencil pass through the same point at infinity determined by the direction of the line ax+by=0.
This interpretation of the equation c=0, makes all possible triples (a,b,c) which are different from (0,0,0)
to admissible triples for coefficients of lines.
Analogous is the procedure by which the impossible triple (a,b,c) is incorporated into the triples expressing
lines in line-coordinates. The "impossible" triple identifies the line at infinity:
                                                                        af₁(x,y)+bf₂(x,y)+cf₃(x,y) = 0.
An arbitrary line which in line coordinates is written:
                                                                        k₁f₁(x,y) + k₂f₂(x,y) + k₃f₃(x,y) = 0,
defines a pencil of parallel lines described by an equation of the form (omitted x,y):
(6)                                                                   ( k₁f₁+k₂f₂+k₃f₃) + k(af₁+bf₂+cf₃) = 0.

Bibliography

[Carnoy] Carnoy, Joseph. Geometrie Analytique Paris, Gauthier-Villars

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