How do I compute the linear index of a 3D coordinate and vice versa?

Question

If I have a point (x, y z), how do I find the linear index, i for that point? My numbering scheme would be (0,0,0) is 0, (1, 0, 0) is 1, . . ., (0, 1, 0) is the max-x-dimension, .... Also, if I have a linear coordinate, i, how do I find (x, y, z)? I can't seem to find this on google, all the results are filled with other irrelevant stuff. Thank you!

Answer 1

There are a few ways to map a 3d coordinate to a single number. Here's one way.

some function f(x,y,z) gives the linear index of coordinate(x,y,z). It has some constants a,b,c,d which we want to derive so we can write a useful conversion function.

f(x,y,z) = a*x + b*y + c*z + d

You've specified that (0,0,0) maps to 0. So:

f(0,0,0) = a*0 + b*0 + c*0 + d = 0
d = 0
f(x,y,z) = a*x + b*y + c*z

That's d solved. You've specified that (1,0,0) maps to 1. So:

f(1,0,0) = a*1 + b*0 + c*0 = 1
a = 1
f(x,y,z) = x + b*y + c*z

That's a solved. Let's arbitrarily decide that the next highest number after (MAX_X, 0, 0) is (0,1,0).

f(MAX_X, 0, 0) = MAX_X
f(0, 1, 0) = 0 + b*1 + c*0 = MAX_X + 1
b = MAX_X + 1
f(x,y,z) = x + (MAX_X + 1)*y + c*z

That's b solved. Let's arbitrarily decide that the next highest number after (MAX_X, MAX_Y, 0) is (0,0,1).

f(MAX_X, MAX_Y, 0) = MAX_X + MAX_Y * (MAX_X + 1)
f(0,0,1) = 0 + (MAX_X + 1) * 0  + c*1 = MAX_X + MAX_Y * (MAX_X + 1) + 1
c = MAX_X + MAX_Y * (MAX_X + 1) + 1
c = (MAX_X + 1) + MAX_Y * (MAX_X + 1)
c = (MAX_X + 1) * (MAX_Y + 1)

now that we know a, b, c, and d, we can write your function as follows:

function linearIndexFromCoordinate(x,y,z, max_x, max_y){
    a = 1
    b = max_x + 1
    c = (max_x + 1) * (max_y + 1)
    d = 0
    return a*x + b*y + c*z + d
}

You can get the coordinate from the linear index by similar logic. I have a truly marvelous demonstration of this, which this page is too small to contain. So I'll skip the math lecture and just give you the final method.

function coordinateFromLinearIndex(idx, max_x, max_y){
    x =  idx % (max_x+1)
    idx /= (max_x+1)
    y = idx % (max_y+1)
    idx /= (max_y+1)
    z = idx
    return (x,y,z)
}

Answer 2

If you have no upper limit on the coordinates, you can number them from origo and outwards. Layer by layer.

(0,0,0) -> 0
(0,0,1) -> 1
(0,1,0) -> 2
(1,0,0) -> 3
(0,0,2) -> 4
   :       :
(a,b,c) -> (a+b+c)·(a+b+c+1)·(a+b+c+2)/6 + (a+b)·(a+b+1)/2 + a

The inverse is harder, since you would have to solve a 3rd degree polynomial.

m1 = InverseTetrahedralNumber(n)
m2 = InverseTriangularNumber(n - Tetra(m1))
a = n - Tetra(m1) - Tri(m2)
b = m2 - a
c = m1 - m2

where

InverseTetrahedralNumber(n) = { x ∈ ℕ | Tetra(n) ≤ x < Tetra(n+1) } 
Tetra(n) = n·(n+1)·(n+2)/6 
InverseTriangularNumber(n) = { x ∈ ℕ | Tri(n) ≤ x < Tri(n+1) } 
Tri(n) = n·(n+1)/2

InverseTetrahedralNumber(n) could either be calculated from the large analytic solution, or searched for with some numeric method.

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Here is my attempt at an algebraic solution (javascript). I am using the substitutions p = a+b+c, q = a+b, r = a to simplify the equations.

function index(a,b,c) {
    var r = a;
    var q = r + b;
    var p = q + c;
    return (p*(p+1)*(p+2) + 3*q*(q+1) + 6*r)/6;
}

function solve(n) {
    if (n <= 0) {
        return [0,0,0];
    }

    var sqrt = Math.sqrt;
    var cbrt = function (x) { return Math.pow(x,1.0/3); };

    var X = sqrt(729*n*n - 3);
    var Y = cbrt(81*n + 3*X);
    var p = Math.floor((Y*(Y-3)+3)/(Y*3));
    if ((p+1)*(p+2)*(p+3) <= n*6) p++;
    var pp = p*(p+1)*(p+2);

    var Z = sqrt(72*n+9-12*pp);
    var q = Math.floor((Z-3)/6);
    if (pp + (q+1)*(q+2)*3 <= n*6) q++;
    var qq = q*(q+1);

    var r = Math.floor((6*n-pp-3*qq)/6);
    if (pp + qq*3 + r*6 < n*6) r++;

    return [r, q - r, p - q];
}