Euclidean space and Euclidean geometry

Question

If we have a Euclidean space $\mathbb{E}^2$, how can we define the Euclidean geometry,i.e. how to determine point,line,or some other things on it?

Answer 1

In my opinion, the most convenient point of view is like this:

Start with a 2 dimensional affine space, that is a set $E^2$ with a faithful and transitive action of a 2 dimensional real vector space $V$. That gives you a space to work with: by definition a point is just an element of the set $E^2$, a line through point $p \in E^2$ with a direction vector $v \in V$ is by definition the set of points $l(p,v)= \{q \in E^2 : q=p+tv, \ t \in \mathbb{R} \} $.

To measure angles and distances we define a non-degenerate symmetric bilinear form called "scalar product" $( \cdot, \cdot): V \times V \to \mathbb{R} $. For example to measure the angle between vectors $v,w \in V$ you have to compute $ \angle(v,w) = arccos\frac{(v,w)}{\sqrt{(v,v)}\sqrt{(w,w)}}$, the distance between points $p$ and $q$ is just the norm of a unique vector $v \in V$ connecting those points - $dist(p,q) = \sqrt{(p-q,p-q)}$.

Then you can develop all standard machinery of Euclidean geometry: define transformations, preserving the scalar product; define areas of elementary figures by introducing skew-symmetric bilinear form (which happens to be a determinant), etc.

You can consult the textbook by M. Audin or M. Berger, also have a look at Geometric Algebra by E. Artin for foundations of such approach.

Answer 2

If by $\mathbb E^2$ you mean the set $\{(x,y)\mid x,y\in \mathbb R\}$ together with the usual vector space structure and the standard inner product, then the geometry is given by the elements of $\mathbb E ^2$ being the points and lines being all subsets of the form $L_{(u,v)}=\{u+\alpha v\mid \alpha \in \mathbb R\}$, where $u,v$ are arbitrary vectors (note though that not all choices of these vectors give rise to distinct lines).

Answer 3

It depends on what you mean by Euclidean geometry.

Some mathematicians say that Euclidean geometry is a branch of mathematics that studies Euclidean transformations (a group of transformations that leave the Euclidean metric unchanged) on a space isomorphic to $\mathbb{R}^2$ (I mean isomorphic in the category of linear/vector spaces).

In that case, complex numbers $\mathbb{C}$ provide a nice way to do planar Euclidean geometry. The points will be complex numbers, lines will be formed as linear equations like $z=at+b$ where $z,a,b \in \mathbb{C}$ and $t \in \mathbb{R}$, the plane will be the $\mathbb{C}$ itself. Euclidean transformations like rotations, translations and reflections can be done using complex numbers in a nice simple way.