# Differences and similarities between Euclidean and Minkowski geometry

I am trying to get my head around the differences and similarities between Euclidean and Minkowski plane geometry.

AS far as I understand it they are both affine geometries meaning the parallel postulate holds in both geometries.

but what is the differnce between them?

Coxeter's "Non euclidean geometry" writes that (as i understand it it) has to do how many points at infinity a line has :

if the ideal point is singular (elliptical polarity) gives Euclidean geometry while if there are an infinite number of ideal points (hyperbolic polarity) it gives Minkowski geometry

But that makes it no clearer for me either, what is that polarity thing?

is there not an easier way to understand there differeces?

Is there no difference in axioms or so?

• The Minkowski plane is just the Minkowski space of dimension $2$. – Alex M. Jul 12 '15 at 13:57
• By "Minkowski space of dimension $n$" one understands the vector space $\Bbb R ^n$ endowed with the billinear form $< (v_1, v_2, \dots, v_n), (w_1, w_2, \dots, w_n)> = v_1 w_1 - v_2 w_2 - \dots - v_n w_n$. A pseudo-distance can be defined as $d(v, w) = \sqrt {|<v,w>|}$. For $n=2$ you get the Minkowski plane. The reason why the two concepts seem different is that the Minkowski plane is mostly used for considerations related to the axiomatic foundations of geometry, while people use Minkowski spaces of dimension $n>2$ mostly for considerations related to the special theory of relativity. – Alex M. Jul 13 '15 at 19:45