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?