Ideal points
The properties of incidence in the euclidean plane show a lack of symmetry,
for while two distinct points always define a line, two distinct lines
either define a point
or are parallel.
To achieve symmetry, we introduce additional points, called
ideal points for the plane.
Note that these are purely abstractions, they do not correspond to points in a picture.
It may be useful to think of the plane viewed at an angle,
then the ideal line corresponds to the horizon.
Definitions
The ideal line for the euclidean plane E is the set
{â : a is a direction in E}.
The elements of the set are the ideal points for E.
The extended plane Ê
consists of E, together with the ideal points for E.
A direction in E may be defined by a line through O, or as a class of parallel lines.
We must now define lines and incidence in Ê.
Definitions
If L is a line on E, then the extended line, L^{+},
consists of L, together with the ideal point associated with the direction of L.
The lines of Ê are the ideal line and the extended lines
L^{+} for L on E.
