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Given that the three fundemental geometries are euclidean geometry (zero curvature), riemannian geometry (positive curvature) and lobachevskian geometry (negative curvature), I am curious as to what the link is between these curvatures and the conic sections of an ellipse, an hyperbole and a circle.

I have seen euclidean geometry be called parabolic geometry, riemannian geometry be called elliptical geometry and lobachevskian geometry be called hyperbolic geometry but I have not found any further explanation. How can the 2D shapes be mathematically linked to the 3D surfaces? Why is a parabola associated with flat space while a hyperbole is associated with a saddle shaped 3D space and an ellipse is associated with spherical space?

I have tried to look at curvature for a connection, however, I have only found explanations of curvature and Gaussian curvature seperately but not anything connecting the curvature of 3D space with 2D space.

I also read that gaussian curvature = 1 - eccentricity. This would theoretically make sense as for example a parabola has an eccentricity of 1 giving a curvature of zero which would relate it to flat space. However, I can only imagine this being true for some special cases as I have struggled to find a published explenation. Is there any proof for this formula?

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  • $\begingroup$ The currently unanswered question Parabolic, Hyperbolic, Elliptic appears related. I myself once asked about the term “elliptic” in particular. If I understood the last point in the answer to that correctly, elliptic has a greek origin related to deficit while hyperbolic would indicate surplus. Both could relate to curvature or interior angle, but the relationship to the conic sections of these names would be rather loose. $\endgroup$ – MvG Sep 22 '18 at 15:18
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AFAIK these names come from far-fetched analogies, and other meanings of these three Greek words, rather than any specific similarities to the similarly named curves.

The name "elliptic geometry" is especially confusing to people who hear it for the first time, because it sounds like it would describe ellipsoids, while it actually deals with perfect spheres. At least it makes some sense if you consider circles/spheres to be special cases of ellipses/ellipsoids (but why not name based on the special case then?).

The name "hyperbolic geometry" feels quite appropriate though if you think of the Minkowski hyperboloid model, and also as you would use hyperbolic functions (sinh, cosh) for most distance-related computations in this geometry.

One quite far-fetched relation between Euclidean geometry and a shape of parabola is that horospheres in Minkowski hyperboloid model are represented as paraboloids, and the geometry on a horosphere is Euclidean.

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