# Euclidean space and Euclidean geometry

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?

• Are you looking for these creatures from differential geometry point of views? Or you are sking about E.G. exactly as we have in High school texts? – mrs Jul 22 '14 at 8:16
• @B.S. differential geometry maybe more likely. – 89085731 Jul 22 '14 at 8:17
• en.wikipedia.org/wiki/Foundations_of_geometry – Yves Daoust Jul 22 '14 at 8:52

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.

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).

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.