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Referring to: Curvature of geodesic circles on surface with constant curvature,

Is it possible to combine further the last two of the three equations in the link given above into a single ODE / PDE to obtain constant geodesic curvature = 1/b?

Aim is to get an equation for u = constant "circles" or loops on surface of constant Gauss curvature K = 1/a^2, ( b not equal to a ) in geodesic polar coordinates in general.

Or, is the last equation alone adequate to define the boundary for all constant K surfaces?

I attempted to numerically find a solution in a procedure that computes constant geodesic curvature in case of constant K surfaces of revolution, positive and negative K, am attaching what might serve as a representative image:

http://i58.tinypic.com/2mm7o06.jpg

( Found an error after uploading: T is actually the radius of curvature = 1/kg).

" J-circle " is nomenclature from an old Russian book that I cannot readily find again.

However, I look for a general formulation that removes restriction on surfaces of revolution.

Regards

I also like to know if the line kg= const on K = const surface is always a closed loop/ oval.

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  • $\begingroup$ Please clarify the question. Combine what? As it stands, the readers have to look at two pages at once to figure out what you are asking about. (And at least to me, even that did not help.) $\endgroup$ – user147263 Aug 10 '14 at 5:44
  • $\begingroup$ @900sit-upsaday. Combine two inputs of constant kg and constant gauss curvature K to obtain one resulting equation containing these two scalar quantities. $\endgroup$ – Narasimham Aug 21 '14 at 19:58

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