The lines define a triangle t = A

Further it is assumed that the lines are in

build a unit-length vector ( a

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

The numbers (k

defined by the three lines {f

not true. Not every triple (k

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

coordinates

(1) N = MM

The (i,j)-entry of the symmetric matrix N is an interesting inner product plus a correction term:

(2) n

With

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) <

where k is the other index of the three than (i,j). Thus the matrix N has the form:

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

The determinant of the above matrix Q is easily found to be zero:

|Q| = 1-2cos(A

The two-dimensional vectors (a,b) express the coordinates of the normal

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(

relations in terms of trilinear coordinates.

See for example [Carnoy, pp. 76-89] who uses the notation {k

must be excluded since they define always a constant:

(4) af

Here D is the area of the triangle of reference and the equality is valid because for every point (x,y), the value

f

Thus, equation (4) is satisfied for

the line coordinates (k

Thus in line-coordinates it can be assumed from a line that it has no

cartesian equation ax+by+c=0.

through x'=x/z, y'=y/z one obtains the

are characterized by the equation z=1. The equation z=0 describes the

points that complements the euclidean plane and turns it to a projective plane.

Analogous is the interpretation of the equation (4). It characterizes all

(5) af

characterizes though the points of 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

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

af

An arbitrary line which in line coordinates is written:

k

defines a pencil of

(6) ( k

Trilinears.html

Produced with EucliDraw© |