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Projective Geometry

Projective geometry is the study of geometrical properties that remain unchanged by “central projection”, which is essentially what happens when you look at parallel elements of a high building like a skyscraper. In projective geometry, one never measures a distance between two points or the angle between two lines. Thus, projective geometry deals with triangles, quandrangles, etc., but never with right-angled triangles, parallelograms, like it is the case in Euclidean geometry. All constructions in projective geometry are not made with a compass and a ruler, but only with a ruler. Despite – or perhaps because of – these simplifications, projective geometry comes up with some theorems, which are generalizations of the theorems found in the Euclidean geometry. For instance, circles, ellipses and parables of Euclidean geometry are special cases of conics in projective geometry. The theory of conics is beautiful in itself and provides a natural introduction to algebraic geometry.

People, who are familiar with the Euclidean geometry regard it as an obvious fact that two parallel lines in a plane never meet in a point, no matter how far we extend them. Surprisingly, in projective geometry, this is not the case. The best possible advice to a student of projective geometry who is new to this subject is to set aside all geometric knowledge methods to work with geometry (s)he is likely to have previously acquired. The best approach to projective geometry is to use its own definitions, axioms and their consequences.