Any good books on non-Euclidean geometry?

[anachin6000]

As the title implies, I'm looking for books on non-euclidean geometry. I'm not looking for very advanced thing, more on some book with a good introduction to this topic.

 

[The Bill]

I like Modern Geometries by Michael Henle. I've only read the first edition, but a professor I know taught out of the second edition and recommends that one as well.

 

[mathwonk]

there are a lot of non euclidean geometries. restricting to 2 dimensional geometry, the euclidean one is distinguished by being geometry on a flat surface. non euclidean geometries are geometry on various curved surfaces. surfaces can curve in basically two ways, like a sphere (positive curvature) or like a saddle (negative curvature. moreover the curvature can either be constant everywhere or can vary. there are thus essentially three geometries of constant curvature, euclidean geometry of zero curvature, spherical geometry of curvature 1, and hyperbolic geometry of curvature -1. certain of euclid's axioms hold in these geometries and others do not, and assuming which ones should hold cause some people to regard only the hyperbolic geometry of curvature -1 as non euclidean geometry. more generally one can consider geometry on surfaces of varying curvature, or "riemannian geometry.

One can also consider geometries on surfaces that are not complete (some curves that start out like lines do not continue forever but circle back on themselves), giving rise to geometries that are locally euclidean but where some global axioms fail. one example is geometry on a cylinder.

as an introduction to spherical, locally euclidean, and a little hyperbolic geometry i like very much the book Geometries and Groups by Shafarevich and Nikulin. another excellent book is stillwell's geometry of surfaces, and if you want the original book on non euclidean geometry you can take a look at euclides vindicatus by saccheri. if you just want to learn some geometry from one of the all time great geometers, you cannot go wrong with any portion of "geometry and the imagination" by hilbert and cohn-vossen.