Equivalently assume that the three lines define the sides of a triangle. In principle all elements and properties of the triangle can be deduced from the coefficients of the three lines defining the triangle:

(0) a

The vertices {A,B,C} of the triangle are opposite to the lines with indices {1,2,3}. The equations:

(1) a

represent the line in standard homogeneous coordinates which setting z=1 give equations (0) in cartesian coordinates.

Assume further that the line equations are in

f

give the signed distance of X(x,y) from the corresponding line defined by f

The first remark is then that three numbers {f

Thus, for every X (see Trilinears.html ):

af

where D is the area of the triangle and {a,b,c} the side-lengths of the triangle. It follows also that

are the absolute barycentric coordinates of the point X with respect to the triangle (see BarycentricCoordinates.html ).

The functions f

The map is linear and in standard coordinates it is described by the matrix (see BarycentricsFormulas.html ):

The area D of the triangle satisfying

we see easily that the inverse matrix is given by

From this follows easily that matrix M maps:

Here h

The meaning of M is also examined in the file BarycentricsFormulas.html . From the discussion there it follows that

From this follows the meaning of the determinant of the trilinears (U

Here the rows of the first matrix from left are the coefficients of three arbitrary lines (not necessarily in normal form),

the columns of the second matrix are the coordinates of the vertices of the resulting triangle. First column represents the vertex opposite to the first line etc.. The matrix equation expresses the fact that each vertex is the intersection of two of the given lines. Looking the solution (x

in a formal way we see that the expression ax

Similar forms for the other expressions in the above matrix equation lead to the equality [Castelnuovo p. 195]:

The three numbers (u,v,w) are the

In the case the lines are normalized the right sides are the ratios of distances of the vertices {(x

Some interesting questions arise, as for example which is the envelope of a set of lines satisfying a linear relation among their line coordinates (or equivalently their coefficients)? The same question can be extended to a higher degree relation among the line coordinates.

The answer is in the realm of projective geometry. In the case of a linear relation ur+vs+wt=0 we obtain a similar relation between the coefficients kr'+ls'+mt'=0 meaning that all lines satisfying such a relation pass through a common point. More generally the answer results from the duality of the projective plane P

More on this aspect of the set of three lines can be found in LineCoordinates.html .

f(x,y,z) = (ax+by+cz)(a'x+b'y+c'z)(a''x+b''y+c''z) = 0.

The above figure displays two level curves f(x,y,z)=0.27 (purple) and f(x,y,z)= -0.53 (red) of this cubic function. An interesting problem is to find cubics passing from as much as possible remarkable points of the triangle.

BarycentricCoordinates.html

BarycentricsFormulas.html

Duality.html

LineCoordinates.html

[Castelnuovo] Castelnuovo, G.

[SalmonConics] Salmon, G.

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