Projective Plane - Vector Space Construction - Desargues' Theorem and Desarguesian Planes

Desargues' Theorem and Desarguesian Planes

The theorem of Desargues is universally valid in a projective plane if and only if the plane can be constructed from a 3 dimensional vector space over a skewfield as above. These planes are called Desarguesian planes, named after GĂ©rard Desargues. The real (or complex) projective plane and the projective plane of order 3 given above are examples of Desarguesian projective planes. The projective planes which can not be constructed in this manner are called non-Desarguesian planes, and the Moulton plane given above is an example of one. The PG(2,K) notation is reserved for the Desarguesian planes.

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