# Definition of constant-curvature curve embedded on an Ellipsoid of revolution

I am interested in identifying a type of curve so I can do literature review on it.

What is the name of a curve embedded on an ellipsoid of revolution in which the curvature of the embedded curve is constant? Or where might I find such a curve described or analyzed?

Background: On the plane, one of the characteristics of an arc (segment of a circle) is that curvature does not change as one traverses the length of the arc. In fields such as surveying, many roadway and railway curves are implemented as arcs. The radiuses of these are usually small relative to the size of the earth, so it is fine that planar arc segments are used for this. Errors are adjusted for at survey-project boundaries.

But for for long alignments (say a river, a mountain range, or an ice crevice), if planar curves are used, the small errors accumulate. It would be desirable to embed the alignment on the ellipsoid of the Earth. Similar issues present when approximating craters on any ellipsoidal body such as the Moon.

I am interested in finding the mathematical definition of a curve embedded on the ellipsoid with a center at any location in which the curve is defined as having a constant Degree of Curvature in the embedded context.

In other words, I want to take this one characteristic of planar circles and arc segments, constant curvature, and figure out what this is on the ellipsoid. Just the name of it, and a reference to its description if anyone happens to know it.

This questions is a follow-on to How to define small circle on an ellipsoid, which I now see was not formed as well as it should have been.

• Do you really need this for an ellipsoid? The Earth might be close enough to a sphere for your purposes. (Just curious.) – Ethan Bolker Mar 9 at 22:53
• @EthanBolker, yes. If a sphere were close enough for my purposes I would still be designing roads to make other people rich. – philologon Mar 9 at 23:20
• You have to use the so-called "Darboux frame" with the two curvatures which are $k_g$, the geodesic curvature and $k_n$, the normal curvature ; see for example web.cs.iastate.edu/~cs577/handouts/surface-curvature.pdf – Jean Marie Mar 9 at 23:43
• Your issue could be considered in terms of "oblate spheroidal coordinates" en.wikipedia.org/wiki/Oblate_spheroidal_coordinates – Jean Marie Mar 10 at 8:28
• I think the answer is not easy. You can find here some ideas, even if in that case constant geodesic curvature was required. – Aretino Mar 10 at 22:00