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Is there already name for the generalization of Clothoids to curves on smooth manifolds, i.e. where the curve's curvature depends linearly on the curve's length-parameter?

In the euclidean plane Clothoids are a suitable idealized model for the trajectory of vehicles moving at constant speed while the steering wheel is also rotated at constant speed and I wonder if there are also idealized models for that kind of driving on smooth surfaces.

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Since this is a reference request: in this article, Arroyo, Barros, and Garay just call what you are describing a Cornu spiral without much fanfare. ("Cornu spiral" or "Euler spiral" are other common names for the clothoid.)

In this report/blog post Jonah Miller speaks about the analogue of the Euler spiral on the sphere (he arrives at those spirals from a different point of view).

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  • $\begingroup$ Nice reference! I was a bit hesitant to "reuse" the term "spiral" because the curve might not look like one, but defining spirals via properties of curvature is also convincing. B.t.w.: elastica is yet another name for Clothoids. $\endgroup$ – Manfred Weis May 22 '17 at 16:09

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