Projective plane visualization
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The document discusses visualizing the real projective plane by extending the real affine plane to include ideal points and lines. Parallel lines in the affine plane are said to meet at an ideal point on the ideal line outside of the plane. The resulting configuration, with both real and ideal points and lines, is called the real projective plane. Key characteristics include that it contains the affine plane plus the ideal line, and has an infinite number of points and lines.
Projective plane visualization
- 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