This is a super interesting discussion for a lot of reasons, and I think what I’m going to come back to is that, the rule book says things that are simply not true.
As an example, canter pirouettes are not physically done the way they have been described for eternity - ie with photographic evidence we know that even the best ones will have two hind feet on the ground and two front feet off the ground during the movement. You have to kind of take it in spirit, which is that it should ‘feel’ like a three beat canter when you watch it at live speed even if it’s not when measured at individual images of 1/30 of a second.
So I think the truth is:
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a 20 m circle is ridden with the spine of the horse coming over X, but with the outer hoofprints tracking at around 19.7 m. The spine cannot do a 20 m circle without a hoof leaving the arena resulting in elimination. The end effect is slightly elliptical but no one will see it.
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a 10 meter circle is done with the interior points based on the spine and the rail at the rail. Thus, the outer hoofprints slightly exceed 10 m, but the spine is exactly 10 m.
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in general, a half circle of any type puts you on the relevant line with the spine down the center of that line. So a 20 m half circle puts you on the track if done at the end of the ring, but puts you on to X if begun in a short arena from A or C. A 10 m half circle from the track goes to the centerline, and vice versa. A 15 m half circle goes from the track to the quarterline.
You know these things are true because of the 20 m figure 8 and serpentines, and the 10 m figure 8 and serpentines.
That this means the math isn’t quite right is okay, because horse people aren’t math people; we’re more science people. In this case, the numbers are labels rather than measurements. Saying 19.7m circles is clumsy and so we are rounding to the nearest meter. And for horses, that’s perfectly fine, because the judges don’t have the tools to measure to the decimeter nor is that level of precision really essential to the quality of the movement.
I applaud the OP for being willing to stand up and ask this question.
