Question

I have two points (x₁, y₁) and (x₂,y₂) which represent the location of two entities in my space. I calculate the Euclidian distance between them using Pythagoras’ theorem and everything is wonderful. However, if my space becomes finite, I want to define a new shortest distance between the points that “wraps around” the seams of the map. For example, if I have point A as (10, 10) and point B as (90,10), and my map is 100 units wide, I’d like to calculate the distance between A and B as 20 (out the right edge of the map and back into the left edge), instead of 80, which is the normal Euclidian distance.

I think my issue is that I’m using a coordinate system that isn’t quite right for what I’m trying to do, and that really my flat square map is more of a seamless doughnut shape. Any suggestions for how to implement a system of this nature and convert back and forth from Cartesian coordinates would be appreciated too!

Answer 1

Toroidal plane? Okay, I’ll bite.

var raw_dx = Math.abs(x2 - x1);
var raw_dy = Math.abs(y2 - y1);

var dx = (raw_dx < (xmax / 2)) ? raw_dx : xmax - raw_dx;
var dy = (raw_dy < (ymax / 2)) ? raw_dy : ymax - raw_dy;

var l2dist = Math.sqrt((dx * dx) + (dy * dy));

There’s a correspondence here between the rollover behavior of your x and y coordinates and the rollover behavior of signed integers represented using the base’s complement representation in the method of complements.

If your coordinate bounds map exactly to the bounds of a binary integer type supported by your language, you can take advantage of the two’s complement representation used by nearly all current machines by simply performing the subtraction directly, ignoring overflow and reinterpreting the result as a signed value of the same size as the original coordinate. In the general case, you’re not going to be that lucky, so the above dance with abs, compare and subtract is required.