<p>As I said in the comments, this is indeed called <em>torus</em> or <em>toroidal space</em> when it comes to the topology. Even if the images suggest something 3 dimensional, this is just a visualization of the embedding of such a space in $\mathbb R^3$.</p>

<p>Regarding the distance between two points, I think you mean following: 
Just consider the coordinates $p=(x_1,y_1)$ and $q=(x_2,y_2)$ in the $[0,X)\times [0,Y) \simeq (\mathbb R / X\mathbb Z)\times (\mathbb R / Y\mathbb Z)$ square. Then define all the translations $q_i$ of $q$ :</p>

<p>$$
\begin{align*}
q_0 &amp;= (x_2\hphantom{+X\,\,\,},y_2\hphantom{+X\,\,\,})\\
q_1 &amp;= (x_2+X,y_2+Y)\\
q_2 &amp;= (x_2\hphantom{+X\,\,\,},y_2+Y)\\
q_3 &amp;= (x_2-X,y_2+Y)\\
q_4 &amp;= (x_2-X,y_2\hphantom{+X\,\,\,})\\
q_5 &amp;= (x_2-X,y_2-Y)\\
q_6 &amp;= (x_2\hphantom{+X\,\,\,},y_2-Y)\\
q_7 &amp;= (x_2+X,y_2-Y)\\
q_8 &amp;= (x_2+X,y_2\hphantom{+X\,\,\,})
\end{align*}$$</p>

<p>Then I think the distance you're looking for is</p>

<p>$$d(p,q) = \min_{i=0}^8 \Vert p-q_i \Vert$$</p>

CC BY-SA 3.0
2017-05-19T08:56:12.097Z
2017-05-19T09:01:19.537Z
2017-05-19T09:01:19.537Z
flawr
<p>I have a 2D space of dimensions (X,Y) with toroidal properties :
In this space, when a point A(x,y) change it's position such as x is getting > X or &lt; 0, then x becomes (x % X).
So when the point reach the bottom of the space, it get back to the top.
And the same for y coordinate.</p>

<p>What is the name of such a space ?</p>

<p>I supposed it was toroidal space, but I can't find any resources with such terms.</p>

CC BY-SA 3.0
2017-05-18T15:37:40.340Z
Is [wikipedia](https://en.wikipedia.org/wiki/Torus) not enough?
flawr
CC BY-SA 3.0
2017-05-19T08:28:19.957Z
Not really, because it describe everything in matter of 3+ dimensions, whereas I would like properties in the 2 dimension space. For example, I would like a formulae to calculate the distance minimal distance between two points.  
I found a way to do it but I'd like more mathematical insights, because I think it would be better and faster.
mpiffault
CC BY-SA 3.0
2017-05-19T08:42:09.070Z
You just see the embeddings of a torroidal space in 3d, but the space itself is 2d. *In the field of topology, a torus is any topological space that is topologically equivalent to a torus.* So a rectangle with opposing sides "glued" together is a toroidal space.
flawr
<geometry><terminology>
What is the name of toroidal 2D space
mpiffault
# What is the name of toroidal 2D space

(Asked by: username_0 on 2017-05-18)

I have a 2D space of dimensions (X,Y) with toroidal properties :
In this space, when a point A(x,y) change it's position such as x is getting > X or < 0, then x becomes (x % X).
So when the point reach the bottom of the space, it get back to the top.
And the same for y coordinate.

What is the name of such a space ?

I supposed it was toroidal space, but I can't find any resources with such terms.

**Tags:** geometry, terminology

> **Comments:**
> **username_1**: Is [wikipedia](https://en.wikipedia.org/wiki/Torus) not enough?  
> **username_0**: Not really, because it describe everything in matter of 3+ dimensions, whereas I would like properties in the 2 dimension space. For example, I would like a formulae to calculate the distance minimal distance between two points.   I found a way to do it but I'd like more mathematical insights, because I think it would be better and faster.  
> **username_1**: You just see the embeddings of a torroidal space in 3d, but the space itself is 2d. *In the field of topology, a torus is any topological space that is topologically equivalent to a torus.* So a rectangle with opposing sides "glued" together is a toroidal space.  

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## 1 answers

### Answer by username_1 on 2017-05-19. Score: 4 | Accepted answer

As I said in the comments, this is indeed called *torus* or *toroidal space* when it comes to the topology. Even if the images suggest something 3 dimensional, this is just a visualization of the embedding of such a space in $\mathbb R^3$.

Regarding the distance between two points, I think you mean following:
Just consider the coordinates $p=(x\_1,y\_1)$ and $q=(x\_2,y\_2)$ in the $[0,X)\times [0,Y) \simeq (\mathbb R / X\mathbb Z)\times (\mathbb R / Y\mathbb Z)$ square. Then define all the translations $q\_i$ of $q$ :

$$
\begin{align\*}
q\_0 &= (x\_2\hphantom{+X\,\,\,},y\_2\hphantom{+X\,\,\,})\\
q\_1 &= (x\_2+X,y\_2+Y)\\
q\_2 &= (x\_2\hphantom{+X\,\,\,},y\_2+Y)\\
q\_3 &= (x\_2-X,y\_2+Y)\\
q\_4 &= (x\_2-X,y\_2\hphantom{+X\,\,\,})\\
q\_5 &= (x\_2-X,y\_2-Y)\\
q\_6 &= (x\_2\hphantom{+X\,\,\,},y\_2-Y)\\
q\_7 &= (x\_2+X,y\_2-Y)\\
q\_8 &= (x\_2+X,y\_2\hphantom{+X\,\,\,})
\end{align\*}$$

Then I think the distance you're looking for is

$$d(p,q) = \min\_{i=0}^8 \Vert p-q\_i \Vert$$


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