Given the projectivity F, select four points {A

The

Applying F

The

All the elementary matrices represent perspectivities (this subject is examined in the file Elation.html ). This could be used to give an alternative proof of the above property, based on the decomposition of an invertible matrix in elementary ones.

From the above arguments follows, more precisely, that every projectivity is the product of two harmonic perspectivities and two elations. On the other hand, as exposed in ElationDecomposition.html , every elation is a product of two harmonic perspectivities. Thus, every projectivity is a product of harmonic perspectivities, the product involving at most six factors.

Harmonic_Perspectivity.html

Projectivity.html

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