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### Desargues Theorem

Projective geometry was invented by the French mathematician Desargues (1591-1661) (for a biography in French, see http://bib1.ulb.ac.be/coursmath/bio/desargue.htm). One of his theorems is considered to be a cornerstone of the formalism. It states that Two triangles are in perspective from a point if and only if they are in perspective from a line'' (see fig. 2.1):

Theorem: Let A, B, C and A', B, C' be two triangles in the (projective) plane. The lines AA', BB', CC' intersect in a single point if and only if the intersections of corresponding sides (AB, A'B'), (BC, B'C'), (CA, C'A') lie on a single line. The theorem has a clear self duality: given two triplets of lines and defining two triangles, the intersections of the corresponding sides lie on a line if and only if the lines of intersection of the corresponding vertices intersect in a point.

We will give an algebraic proof: Let P be the common intersection of AA', BB', CC'. Hence there are scalars such that: This indicates that the point on the line AB also lies at on the line A'B', and hence corresponds to the intersection of AB and A'B', and similarly for and . But given that the three intersection points are linearly dependent, i.e. collinear. Exercise 2.4   :  The sun (viewed as a point light source) casts on the planar ground the shadow A'B'C' of a triangular roof ABC (see fig. 2.2). Consider a perspective image of all this, and show that it is a Desargueian configuration. To which 3D line does the line of intersections in Desargues theorem correspond? If a further point D in the plane ABC produces a shadow D', show that it is possible to reconstruct the image of D from that of D'.     Next: Hyperplane Transformations Up: Hyperplanes and Duality Previous: The Duality Principle
Bill Triggs
1998-11-13