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Perspectivity

You may drag most of the elements in the interactive.

This, a “Perspectivity”, is the simplest, purely-projective comparison of interpunctual intervals. It all happens in a single plane.

• The red interval is to be compared with the black by way of the orange and green lines incident in point P, through these two intervals' end-points,
       while ...
• The orange interval is to be compared with the green by way of the red and black lines incident in point Q, through these two intervals' end-points.

If the end-points do, in fact, “connect up”, (come to be incident) with P, or with Q,
we can at once say the corresponding intervals are
equivalent or identical.

 They match by incidence, so are projectively indistinguishable. 

Now the Euclidean approach to comparison is exactly the same as the Projective, except that parallel, not central, projection is invoked, and the comparison is expected to be by size.  Go here to see how this is resolved through Perspective, and how parallel projection turns out to be central, after all!
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