How to map square coordinates to a circle?

Questions & Answers
1

Hey.

I have a square 2D minimap. I want to make a circular version of that minimap, so I want to map the original square minimap’s X and Y coordinates onto an inscribed circle.

How does one map a square XY coordinate onto a circle of equal dimensions? The original square coordinates are stored in a Vector2 “pos”, and the new circle coordinates are stored in a Vector2 “newPos”.

I had tried the following function. But I must have made some error, as it’s returning undefined numbers instead of proper coordinates.

    public Vector2 squareToCirclePosition(Vector2 pos)
    {
        Vector2 newPos = new Vector2(0,0);
        newPos.x = pos.x * Mathf.Sqrt(1 - ((Mathf.Pow(pos.y, 2F)) / 2));
        newPos.y = pos.y * Mathf.Sqrt(1 - ((Mathf.Pow(pos.x, 2F)) / 2));
        ...
        return newPos;
    }

Any ideas here? Sorry if I’m clueless at math. And scripting.

Thanks!

2

There’s one main, basic problem here: A square does not have equal dimensions to a circle.

That won’t prevent you from displaying your minimap in a circular area, but you may want to ask yourself a question about what you’re looking to do:

Are you sure you want to convert these distances in this manner? If you try and make use of the entire area of a square, compressed into the shape of a circle, anything not exactly on a cardinal direction on your minimap will be at a skewed relative location because rather than a 1:1 ratio of object position to map position, it will be at a 1:sqrt(xx+yy) ratio, depending on its position around the circle.

It may be more prudent to simply keep track of your minimap in the shape of a circle exclusively, instead. This would provide a (potentially) equidistant range on your minimap without favoring diagonals (not counting a square/rectangular minimap where that’s intended) by basing what’s displayed on it purely on its distance from the player.

If you do wish to proceed with remapping a square to a circle as-is, that is done by dividing the position of the object displayed by the maximum length to reach the edge of the minimap (i.e. an enemy at position (0.2, 0.4) on a -1 to 1 range minimap remaps to (0.1789, 0.3578) by dividing both side by 1.1180, the length of the line found by dividing 1 by the greater of the x or y value)

If you’re absolutely sure you want to do this, remapping the vectors would be done as follows:

Vector2 SquareToCircle(Vector2 input)
{
	if(input.sqrMagnitude == 0)
	{
		return Vector2.zero; // Avoid division by 0
	}
	Vector2 output;
	float multiplier = 1.0f / Mathf.Max(input.x, input.y);
	output = input * multiplier; // Extend the line to the edge of the square
	float divisor = Mathf.Sqrt((output.x * output.x) + (output.y * output.y));
	output = input / divisor; // Contract the original point away from the circle's edge
	return output;
}

Edit: The simplest alternative, if you simply wish to use a circular shape, is to assemble the square map, but then cull contents outside the circular shape.