## Table of Contents

I adapted the answers and plotted the results:

*Ilmari Karonen*’s answer based on *atan2*:

function octant1(vector) { let angle = Math.atan2(vector.y, vector.x); let octant = Math.round(8 * angle / (2*Math.PI) + 8) % 8; return octant; }

*Lars Viklund*’s answer based on *dot products*:

let candidates = []; for (let octant = 0; octant < 8; octant++) { candidates.push({x: Math.cos(Math.PI * octant/4), y: Math.sin(Math.PI * octant/4)}); } function octant2(vector) { function dot(a, b) { return a.x * b.x + a.y * b.y; } let bestOctant = 0, bestDotProduct = 0.0; for (let octant = 0; octant < 8; octant++) { let dotProduct = dot(vector, candidates[octant]); if (dotProduct > bestDotProduct) { bestOctant = octant; bestDotProduct = dotProduct; } } return bestOctant; }

*ChrisC*’s answer based on *complex numbers* (although it seems the same as if you had used vectors):

function octant3(vector) { const quadrantToOctant = [5, 6, 7, 4, 0, 0, 3, 2, 1]; let scale = Math.max(1e-6, Math.max(Math.abs(vector.x), Math.abs(vector.y))); let xDirection = Math.round(vector.x / scale); let yDirection = Math.round(vector.y / scale); let quadrant = (yDirection+1)*3 + (xDirection+1); return quadrantToOctant[quadrant]; }

As pointed out in the comments by *izb*, this fails! **It switches octants at the wrong angle.** It switches when the ratio `vector.x / vector.y`

crosses 0.5, which causes the rounded value go switch from 0 to 1. That’s *arctan(0.5)* which is 26.565°. So it’s clever but it doesn’t quite work like the others. Let’s try to fix it.

With complex numbers, *multiplication* involves rotation. If you start with East and multiply it by 1/8th of a circle rotation (call this M), you end up in the next octant. Each time you multiply, you move one quadrant. So one way to think about this is to ask *how many rotations* does it take to go from East to where we are now? Equivalently, *how many multiplications* does it take?

where we are now V = East * M * M * M * … (N times) where we are now V = East * M^N N = logarithm(base M) of V

So we need a logarithm in complex number space^{[2]}. This involves arctangent. So we are back to arctangents! A logarithm in base M is log(V)/log(M), which means atan2(V.y, V.x) / atan2(M.y, M.x). Since M is 1/8th of a circle this ends up being 8 * atan2(V.y, V.x). This is the same solution as at the top of the page.

## 1 Appendix: drawing code#

All the code for this page is displayed on this page. Here’s the rest:

function fillCanvas(canvas, calculateOctant) { const octantToColor = [ [180, 130, 50], [150, 180, 50], [50, 180, 50], [50, 180, 150], [50, 130, 180], [80, 50, 180], [180, 50, 180], [180, 50, 80], ]; const ctx = canvas.getContext('2d'); const imageData = ctx.getImageData(0, 0, canvas.width, canvas.height); const pixels = imageData.data; for (let x = 0; x < canvas.width; x++) { for (let y = 0; y < canvas.height; y++) { let start = 4 * (y * canvas.width + x); let octant = calculateOctant({x: 3*(x/canvas.width - 0.5), y: 3*(y/canvas.height - 0.5)}); for (let channel = 0; channel < 3; channel++) { pixels[start+channel] = octantToColor[octant][channel]; } pixels[start+3] = 255; } } ctx.putImageData(imageData, 0, 0); } fillCanvas(document.getElementById('octant1'), octant1); fillCanvas(document.getElementById('octant2'), octant2); fillCanvas(document.getElementById('octant3'), octant3);