How does spherical geometry differ from what you are looking for?

See also

How does spherical geometry differ from what you are looking for?

See also

I can do that. However, literature would be nice to refer to in a write-up.

Are you referring to this “r” in the parameterization equations for a half circle?

Also I must keep in mind that I need to imbed this sphere into 4D. It does grow.

I am talking about something like this:

The line element for a one-dimensional circle that is not embedded into anything is:

ds^2 = r^2 \textrm{d}\theta^2 \; ,

The r here is just a parameter that has no meaning of “radius”. There is no radius, and there could not be any radius, as we are in one-dimension. In addition, there is no “radial coordinate”. The only reason people write that parameter down with the suggestive letter r is because when we embed the one-dimensional circle into the **very special** 2-dimensional space R^2, then that r becomes the radius that we know from elementary school geometry.