No perfect circles in space...

[Anomalocaris]

As an example, take a flat surface, make a dimple on it.   Trace an enclosed line, exactly equidistant from some other point on the surface as measured along the surface, and passing through the dimple.   This line forms a perfect circle from the perspective of the surface.

I'm not being intentionally difficult, I'm seriously trying to grasp this.

On a flat surface, the circumference of a circle with a diameter of 10 cm would be 31.4 (ish)cm.

Ok, now we have a flat surface with a dimple.  Let's say the dimple extends 1cm below (or above, doesn't really matter) the rest of the surface. We choose our centre point 5 cm from the dimple, giving us a radius of 5 and a diameter of 10.  But the line marking the circumference of the circle would, when passing through the dimple, go down 1 cm then up 1 cm, making the circumference of this circle 33.4 cm, so C = pi x d wouldn't apply, making this not a perfect circle. 

What am I missing?

Boru

If you define the circle as circumference = pi X D, then it won’t be a circle.  But that relation is normally taken to be the property of a circle inscribed on a flat plane, not the definition what circle is.

But we normally define the circle as a locus of points all of whom are exactly the same distance from a single center point, and forming an enclosed circumference.   Using such a definition the circle on a dimpled surface is a true circle, but it would not have the property of  circumference = pi X D.  For such a circle, it is precisely as you say, the circumference, if measured by tracing along the dimpled surface, would go down and then up, so that is precisely the reason why its circumference would be longer than the circumference of a circle of exactly the same diameter but inscribed on a perfectly flat plane.

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I can't imagine a dimple on a flat surface in anything other than 3D - dimpling the a flat surface changes two dimensions to three, doesn't it?

Boru

True.  But, we live in a 3D world.  If a satellite is in orbit about the Sun (say, by NASA), and the eccentricity of its orbit is 0, then, its orbit is a circle.  Why does the diameter of its orbit times pi not equal the circumference?  And, how can that be visualized?

I don’t know if this is the crux of the issue.  But I would like to point out the orbit of the earth is not quite circular, but instead is a ellipse with a small non-zero eccentricity.  I believe even without any perturbation from any third source of gravity, Relativistic effects would make the earth not repeat the exact same complete elliptical orbit trace time after time.  Instead the earth would behave as if each each elliptical orbit ends just a tiny fraction too early, and next orbit starts slightly early, so the major axis of the ellipse gradually rotate through space, making one complete revolution every few hundred million years.  

I think, (but do not know for certain) that is what the quoted text meant when it said earth’s orbit is 10km shorter than pi X 2 X distance from earth to the sun projected onto 3D space.