A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Robin Stuart
Date: 2017 Feb 7, 08:32 -0800
I have inserted words in square brackets  into your text meant to clarify the way I interpret them.
The key point is in the definition of the CoP.
Circle of equal altitude: a circumference [locus] on the surface of the earth, on every point of which the altitude [or zenithal distance] of a given celestial body is the same at given instant.
As I said before, it not depends on the Earth´s model: sphere, ellipsoid or geoid.
The center of the CoP is the geographical position of the star.
I agree with all this although the exact interpretation of the last sentence needs a little care.
And you wrote:
The distance [along the surface of the Earth] from this center to the CoP is the zenith distance of the celestial body.
No. It is on a sphere but is not in general true.
And you wrote:
Distance: great circle, great ellipsoid, geodesic, ..., the real one.
I’m not sure what you mean by this. Great ellipsoid and geodesic distances are will be different in general and I don’t understand the concept of the “the real one”.
Let’s consider a limiting example. Suppose the GP lies on the equator and prime meridian at 0°N, 0°E and construct an LoP corresponding to a zenithal distance 90°. It’s pretty easy to see that the North Pole and the point 0°N, 90°E both lie on the LoP because the verticals at those points are at 90° to the vertical at the GP (whether we are talking about a sphere or ellipsoid). If I measure the distance along the Earth’s surface from the GP to the North Pole I will get 10,002 km. If I measure the distance on the Earth’s surface from the GP along the equator to the point 0°N, 90°E I will get 10,007.5 km. These geodesic distances from the GP are different even though their zenithal distances are the same.
Is there anything in the above that you disagree with? If so please be identify it specifically. This is a special case of what I was saying. No more, no less.