November 6

Introduction to Projective Geometry using Desargues' Finite Geometry

Gerard Desargues (1593 - 1662) was a French mathematician who gave much of the beginnings of projective geometry. Projective geometry's first formal text, Traité des propriétés des figures by J. V. Poncelet (1788 - 1867), was published in 1822. Interesting fact is that Poncelet did most of his work on this text while in prison in Russia after the Napoleonic wars.

Definitions for the Finite Geometry of Desargues 1. Two triangles are perspective from a point P means that P is that common point joining the lines from the corresponding vertices of the triangles. In the example below, triangle ABC and triangle A'B'C' are prespective from point P. 2 Two triangles are perspective from a line , if the (extended) corresponding sides of the triangles meet at points on line . In the example below, triangle DEF and triangle D'E'F' are perspective from the line , since their corresponding sides, when extended, meet at points X, Y, and Z on line .   3. The line in the finite geometry of Desargues is a polar of the point P if there is no line connecting P and any point on . 4. The point P in the finite geometry of Desargues is a pole of the line if there is no point common to and any line on P.

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Axioms for the Finite Geometry of Desargues 1. There exists at least one point 2. Each point has at least one polar. 3. Every line has at most one pole. 4. Each two distinct points are on at most one line. 5. Every line has exactly three distinct points on it. 6. If a line does not contain a certain point, then there exists a point on both the line and any polar of the point. In symbols, if is a line, P is a point, and P , then and the polar of P intersect.

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Model for the Finite Geometry of Desargues

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Theorems for the Finite Geometry of Desargues 1. Every line has exactly one pole. 2. Every point has exactly one polar.

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Questions for the Finite Geometry of Desargues HINT: Parallel lines exist in this geometry, but their properties are different. sometimes there are more than one line parallel to a given line and sometimes there is only one line parallel. to a given line. For example, in the model above, three different lines can be drawn parallel to line R, C, B through point A', but only one line can be drawn through A' parallel to line A, B, T.

from Modern Geometries by James Smart pp. 28 - 31.

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