[SOLVED] how do I find a coordinates given a number in Spiral pattern?

how do I find a coordinates given a number in Spiral pattern?

given a spiral of numbers in which the numbers are arranged in the form of a triangle, i need to write a function that takes a number and returns the coordinates of this number

           15
           16 14
           17 3  13
           18 4  2  12
           19 5  0> 1  11
           20 6  7  8  9  10
           21 22 23 24 25 26 27 28

for example 17 the result is (-2, 2) in x and y

I already did a similar task, but there the spiral was square and the program received coordinates (x, y) as input and returned a number. in short, there I calculated the size of the square and the offset.

I will be glad for any help

Solution

TL;DR: Working code at the bottom

First consider how many numbers you have in each triangle:

The first contains 3 numbers in each edge
The second contains 6 numbers in each edge
The third contains 9 numbers in each edge

So there is our pattern. Triangle i contains 9i numbers and the before we do anything else we need to divide our number by 9 and find the triangle root of the result (and round it down):

Once you figure out the right triangle, there are three things left to do:

You need to find the starting point

This is trivial as the "last point" of triangle i will always be (2i, i).

You need to find the right edge

You already know that your triangle has i-long edges so by taking the sum of your remainders (from the original divmod and from rooting) and dividing it by 3 you can find the right edge.

You need to find the right point on this edge

This bit is trivial - you have 3 types of edges, horizontal, vertical and diagonal. Depending on the type you have to apply your "final residual value" to the "origin of the edge":

(-r, -r) for the diagonal
(0, r) for the vertical
(r, 0) for the horizontal

Relative to the max point of the previous triangle to get to the right edge you just have to apply a cumulative sum of these transpositions:

(-r, -r) for the diagonal
(-i, -i+r) for the vertical
(-i+r, 0) for the horizontal

Putting it all together

def triangle_root(x):
    return int((8 * x + 1) ** 0.5 - 1) // 2

def spiral(v):
    # Identify the current triangle
    i = min(v, triangle_root(max(0, (v - 1) / 9)) + 1)
    # Compute the coordinates for the max value of this triangle
    xi, yi = 2 * i, i
    # Compute the point index in the current triangle
    # In other words, subtract the previous triangle max
    r = v - 9 * (i - 1) * i // 2
    # Compute the edge length for the current triangle
    length = 3 * max(1, i)
    # Compute the current edge and the location in that edge
    edge, r = divmod(r, length)
    # Apply the relevant transform depending on the edge
    if edge == 1: # vertical
        dx, dy = -length, r - length
    elif edge == 2: # horizontal
        dx, dy = r - length, 0
    else: # diagonal
        dx, dy = -r, -r
    
    return xi + dx, yi + dy