On ellipses, spheres and inverses of rectangles.

We are quite acquainted with the usual geometries common in astronomy, ellipses, spheres, big circles, small circles.....But, I was just thinking of more funny shapes that are found in this system.Let's just start with say the equivalent of an ellipse on a sphere.But oops!!! Spheres never have ellipses, planes always cut them in circles.The first thing that strikes us is projection of an ellipse on a sphere.So maybe we could try redefining our spherical ellipse as the projection of a planar ellipse on a sphere.But how do we do that??We would need a Haumea (an almost ellipsoid dwarf planet in our solar system at about the distance of Pluto http://antwrp.gsfc.nasa.gov/apod/ap080923 ) to be revolving around the Earth, so that it casts an "ellipse" for a shadow on the Earth.But on second thoughts, do we really need an ellipsoid to cast a shadow. We have such a pretty, round, big moon revolving around us, why cann't we use it so as to satisfy our purpose. Its shadow happens to be a cone, and as you have most probably guessed by now, a plane cutting a cone gives us an ELLIPSE, so a sphere cutting a cone might be called a "spherical ellipse". The actual maths of the problem may lead us to some amazing properties of the curve obtained.There are a few other examples I came up with, but would like to have a really great discussion on ideas from others, so just put on your ingenious brain to do some great work and get out more beatiful ways of doing it, or maybe finding out other interesting shapes we find in astronomy.

Viraj Deshpande