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On Infinity

On Measurement

On Metrics

On the Algebra of Measure
        (2) Perspective measure
        (3) Geometric measure
        (4) "Growth" measure

Ceva's Theorem

When a range of points on one line is projected from a point into another range on another line, metric qualities, such as size, or angle, are not preserved from range to range, simply because, for pure projective geometry, there are no such qualities to preserve!

If, however, a standard measure, such as the SRM, is pressed into service, then it is possible to assign numerical sizes, to the intervals between points (or, equivalently, the segments the points define), and then it is found that a ratio of ratios, or "Cross Ratio", of any three such intervals is preserved (i.e. is constant) under projection from range to range—and that this (also in its dual form) is the only metric quality to be so.

For intervals a, b and c on one line projecting into A, B and C respectively on another line, we may, for example, write

[a/(b+c)]/[(a+b)/c] = [A/(B+C)]/[(A+B)/C]

The proof (using Ceva's Theorem) of this very important result is given in the animation alongside.

Concerning Sectors

We have,

Interestingly, for the dual form, concerning sectors of pencils, rather than segments of lines, when evaluating cross ratio, we have not to do with the angles of the sectors, but instead with the sines of those angles—and sines are, of course, ratios of distances ('angle' is actually something of a cuckoo in the geometric nest). So even with rotation, we are still dealing with ratios of ratios of distances.