On a two dimensional grid is there a formula I can use to spiral coordinates in an outward pattern?

Here’s a recipe for finding the coordinates of your position after $n$ steps along the spiral.

It’s simpler to number the positions on the spiral starting at $0$: position $0$ is $\langle 0,0\rangle$, the origin, position $1$ is $\langle 1,0\rangle$, position $2$ is $\langle 1,-1\rangle$, and so on. Using $R,D,L$, and $U$ to indicate steps Right, Down, Left, and Up, respectively, we see the following pattern:

or with exponents to denote repetition, $R^1D^1|L^2U^2\|R^3D^3|L^4U^4\|R^5D^5|L^6U^6\|\dots\;$. I’ll call each $RDLU$ group a block; the first block is the initial $RDLLUU$, and I’ve displayed the first three full blocks above.

Clearly the first $m$ blocks comprise a total of $2\sum_{k=1}^mk=m(m+1)$ steps. It’s also not hard to see that the $k$-th block is $R^{2k+1}D^{2k+1}L^{2k+2}U^{2k+2}$, so that the net effect of the block is to move you one step up and to the left. Since the starting position after $0$ blocks is $\langle 0,0\rangle$, the starting position after $k$ full blocks is $\langle -k,k\rangle$.

Suppose that you’ve taken $n$ steps. There is a unique even integer $2k$ such that $$2k(2k+1)<n\le(2k+2)(2k+3)\;;$$ at this point you’ve gone through $k$ blocks plus an additional $n-2k(2k+1)$ steps. After some straightforward but slightly tedious algebra we find that you’re at

$$\begin{cases} \langle n-4k^2-3k,k\rangle,&\text{if }2k(2k+1)<n\le(2k+1)^2\\ \langle k+1,4k^2+5k+1-n\rangle,&\text{if }(2k+1)^2<n\le 2(k+1)(2k+1)\\ \langle 4k^2+7k+3-n,-k-1\rangle,&\text{if }2(k+1)(2k+1)<n\le4(k+1)^2\\ \langle -k-1,n-4k^2-9k-5\rangle,&\text{if }4(k+1)^2<n\le2(k+1)(2k+3)\;. \end{cases}$$

To find $k$ easily, let $m=\lfloor\sqrt n\rfloor$. If $m$ is odd, $k=\frac12(m-1)$. If $m$ is even, and $n\ge m(m+1)$, then $k=\frac{m}2$; otherwise, $k=\frac{m}2-1$.

Here is some code that finds the $n$-th point in the spiral. Unfortunately it spirals the other way but perhaps it helps anyway.

function spiral(n)
        k=ceil((sqrt(n)-1)/2)
        t=2*k+1
        m=t^2 
        t=t-1
        if n>=m-t then return k-(m-n),-k        else m=m-t end
        if n>=m-t then return -k,-k+(m-n)       else m=m-t end
        if n>=m-t then return -k+(m-n),k else return k,k-(m-n-t) end
end

See http://upload.wikimedia.org/wikipedia/commons/1/1d/Ulam_spiral_howto_all_numbers.svg.

If you are looking for a no-if solution and a formula, I was able to find this one:

$A = ||x| - |y|| + |x| + |y|;$

$R = A^2 + sgn(x + y + 0.1)*(A + x - y) + 1;$

$x, y \in \mathbb{Z}$

$sgn$ — sign function

<?php

$n = 4;
$from = -intval($n / 2) - 1;
$to = -$from + ($n % 2) - 2;

for ($x = $to; $x > $from; $x--) {
    for ($y = $to; $y > $from; $y--) {
		$result = pow((abs(abs($x) - abs($y)) + abs($x) + abs($y)), 2) + abs($x + $y + 0.1) / ($x + $y + 0.1) * (abs(abs($x) - abs($y)) + abs($x) + abs($y) + $x - $y) + 1;
		echo $result . "\t";
    }
    echo "\n";
}

which prints

7   8   9   10  
6   1   2   11  
5   4   3   12  
16  15  14  13

https://repl.it/repls/DarkslategraySteepBluebottle

On a two dimensional grid is there a formula I can use to spiral coordinates in an outward pattern?

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Coordinate Systems

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