Recommend me a geometry book

mikhail

I've had a lot of success applying basic geometry to problems which can be expressed that way over the years. I have a PhD in engineering, but I reckon most of the geometry I know comes from secondary school. I've always compensated with a strong faculty for spatial relations. Recently, I´ve found myself struggling with the limitations of that - I'm working on generalising a very useful 2D result to 4D, and my lack of formal geometry has slowed me down a lot. Even something as simple as co-ordinate transformation of a line in 4D space - I've had to look up how to write the line (I quickly realised that l w + m x + n y + o z + c = 0 defines a volume!), and my intuition struggles with the transformations (it's not easy to picture 4D!)

Can anyone recommend a geometry textbook or books which would suit me? I think something which is simple but rigourous, or which compiles a lot of ideas and tricks for intuitive work would be appropriate (both would be ideal!).

Fremen

Geometry is a very broad term in mathematics. It sounds like you're looking to learn some linear algebra. "Elementary linear algebra" by Anton is quite good, though I'd be surprised if you didn't know that back to front with an engineering PhD.

"Differential geometry" by Kreyszig will probably be pretty good, though considerably more andvanced and more "geometric" than Anton. I haven't actually read it, but I've read another book by the same author and he writes very clearly.

Do Carmo's book on Riemannian geometry is also good, though relatively advanced and possibly more abstract than you need. The idea behind Riemannian geometry is to take an object which looks "locally" like euclidean space, in the same way that a smooth curve looks "locally" like a straight line (when you zoom in very closely), and to work with this "almost euclidean" guy in much the same way as with euclidean space.

mikhail

Fremen wrote: »

Geometry is a very broad term in mathematics. It sounds like you're looking to learn some linear algebra. "Elementary linear algebra" by Anton is quite good, though I'd be surprised if you didn't know that back to front with an engineering PhD.

As you say, I probably know that stuff. I just haven´t applied it or thought about it in the right way.

Thanks for the recommendations.

cavedave

If you can program Turtle geometry is good. It uses logo.

sponsoredwalk

Harold Coxeter wrote Introduction to Geometry which is chock full of stuff.

That book looks a bit advanced though so I'd say using this highly recommended book (on physicsforums.com) along with Euclid's Elements books I-IV as that book asks you to do would more than prepare you for Coxeter.

That is at least the way I'm doing it

If you're looking for Differential Geometry I know of a 5-volume piece de resistance by Michael Spivak.

Just do a bit of online research into these books and be sure before you buy, (assuming it's not just linear algebra that you're talking about)