Projective Geometry

1. Real projective plane: Visualization
Let us consider a trapezium as a paper sheet, which is merely a representative picture of affine plane. (Fig. 1.1)
Real affine plane
Figure. 1.1
Figure. 1.2
Figure. 1.3
Figure. 1.4
Real projective plane
Ideal point
Ideal line
Figure. 1.5
Figure. 1.6
Figure. 1.7
We may sort these ordinary lines in affine plane into parallel classes that consists of a collection of lines parallel to
one another. (see fig. 1.4)
In projective plane, for each such parallel class we invent an ideal point on which these parallel lines do meet (see
Fig. 1.5).
These ideal points don’t lie on the Real Affine Plane so we can put these ideal points outside of the shape that
represents the Real Affine Plane (trapezium). As ordinary points lying on ordinary lines in the usual way, we set that
all lines in a given parallel class pass through a ideal point and ideal line passes through all such ideal points. (see
Fig. 1.7)
The resulting idea is called the Real Projective Plane. It contains real affine plane along with the ideal line.
Things to remember
(1) Real projective plane includes ordinary lines and ideal line
(2) Extended plane is a projective plane
(3) It has infinite number of points and that of lines
(4) It is not a configuration
(5) If two points in real proective plane are given, then line on both is obtained by their cross product
(6) If two lines in real proective plane are given, then point on both is obtained by their cross product
(7) Real projective plane includes real affine plane cum ideal line
(8) Real projective plane includes ordinary points and ideal points