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Projectively Proscribed Items


Projective Geometry is Exclusive GeometryIt preserves Incidence, but not Absolute Measure.In particular, Equality and Isotropy are not preserved, and there is no Axiom of Curvature, In consequence, Parallelity, Rectangularity and Circularity, are not “selfevident” givens. They are meaningless, and in fact undetectable.Scalars, Vectors and Tensors do not work. Solid, Rigid Rotation is not available, 

Slipfree, ‘MeasurePreserving’, Rolling Operations, such as the rolling of a truck wheel on the ground, or printing from a cylindrical plate on to a plane, or beltandpulley transmission, cannot be had from strictly elementary geometric construction. Forms cut by tools such as the twist drill depicted above also cannot be had from strictly axiomatic geometric construction. Figures made by compasses cannot be shown to be circles. The hyperboloid adumbrated above cannot be made by rotation around a line of a plane containing a hyperbola, because that hyperbola is not rigidly preserved during rotation  in other words, the hyperbola is not “imprinted”, or “mounted” like a painting on a wall, on the rotating plane, so cannot be transported by it. 

Nineteenthcentury mathematicians
(Notably, Felix Klein and Arthur Cayley), saw a need to heal this breach between Projective and Euclidean Geometries. 

It is unlikely that just three vanishing points can accommodate all the features of a real building like this one in London, although those indicated above seem to do it, if in a rough, ‘broad brush’ fashion. Perhaps the full set would cluster about these three. 