Editing Homogeneous coordinates (section)

==Trilinear coordinates==
{{Main|Trilinear coordinates}}
Let <math>l</math>, <math>m</math> and <math>n</math> be three lines in the plane and define a set of coordinates<math>X</math>, <math>Y</math> and <math>Z</math> of a point
<math>p</math> as the signed distances from <math>p</math> to these three lines. These are called the ''trilinear coordinates'' of <math>p</math> 
with respect to the triangle whose vertices are the pairwise intersections of the lines. Strictly speaking these are not
homogeneous, since the values of <math>X</math>, <math>Y</math> and <math>Z</math> are determined exactly, not just up to proportionality. There is
a linear relationship between them however, so these coordinates can be made homogeneous by allowing multiples of
{{nowrap|<math>(X,Y,Z)</math>}} to represent the same point. More generally, <math>X</math>, <math>Y</math> and <math>Z</math> can be defined as
constants <math>p</math>, <math>r</math> and <math>q</math> times the distances to <math>l</math>, <math>m</math> and <math>n</math>, resulting in a different system of
homogeneous coordinates with the same triangle of reference. This is, in fact, the most general type of system of
homogeneous coordinates for points in the plane if none of the lines is the line at infinity.<ref>
    {{harvnb|Jones|1912|pp= 452 ff}}</ref>