Summary
We say that a surface has a “rectilinear geodesic circle“ if it contains a straight line segment AB and a point F whose geodesic distance from a variable point of AB is a constant.
The problem of generating such surfaces is solved here by constructing families of cones which behave locally (along a curce) like solutions, and then taking their envelopes. This method generates a class of solutions which depend on an arbitrary curve.
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This is an excerpt from a thesis presented in partial fulfillment of the Ph. D. Requirements of the Mathematics Department of Stanford University. It was supported in part by the Ballistics Research Laboratory of the Ordnance Department, U. S. Army, under Contract No. DA-04-200-ORD-294.
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Garsia, A. On surfaces with a rectilinear geodesic circle. Annali di Matematica 46, 201–213 (1958). https://doi.org/10.1007/BF02412916
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DOI: https://doi.org/10.1007/BF02412916