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.

  • $\begingroup$ Is wikipedia not enough? $\endgroup$
    – flawr
    May 18 '17 at 15:37
  • $\begingroup$ 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. $\endgroup$
    – mpiffault
    May 19 '17 at 8:28
  • $\begingroup$ 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. $\endgroup$
    – flawr
    May 19 '17 at 8:42

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

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

  • $\begingroup$ You're right, I found some good stuff with "toroidal space" after all. And thanks for the algorithm. I think I have an optimization I will test and confirm here. I'll accept your answer after that. $\endgroup$
    – mpiffault
    May 19 '17 at 21:48

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