How to change references of rows and columns to cartesian coordinate?

I generating square grid with inscribed circle. values in circle denoted by 1 and those of square by 0.

import numpy as np
from typing import List, Tuple

def collect(x: int, y: int, sigma: float =3.0) -> List[Tuple[int, int]]:
   """ create a small collection of points in a neighborhood of some point 
   """
   neighborhood = []
   X = int(sigma)
   for i in range(-X, X + 1):
       Y = int(pow(sigma * sigma - i * i, 1/2))
       for j in range(-Y, Y + 1):
           neighborhood.append((x + i, y + j))
   return neighborhood

def plotter(sigma: float =3.0) -> None:
   """ Plot a binary grid """    
   arr = np.zeros([sigma * 2 + 1] * 2)
   points = collect(int(sigma), int(sigma), sigma)
   # flip pixel value if it lies inside (or on) the circle
   for p in points:
       R, C = p
       arr[R][C] = 1
   print(arr)
grid = plotter(10)
print(grid)
print(grid.shape)

The improvement I wanted to do here: Reference to each element to the grid is through grid[R][C], where R is the row and C is the column. How can I convert references of R and C to a cartesian system? For the above case, it should start at (-10,10), where the center point will be (0,0).

what logic I should use to get desired results. and most important when I will call ex. grid[-2][3] it should able to refer the location and get value at that point i.e. 0 or 1.

for x in range(len(grid)):
    for y in range(len(grid[x])):
        if (x > len(grid)/2 and y < len(grid)/2):
            print ("lies in First quadrant")

        elif (x < len(grid)/2 and y >len(grid)/20):
            print ("lies in Second quadrant")

        elif (x < len(grid)/2 and y > len(grid)/2):
            print ("lies in Third quadrant")

        elif (x > len(grid)/2 0 and y < len(grid)/2):
            print ("lies in Fourth quadrant")

        elif (x == len(grid)//2 and y < len(grid)/2):
            print ("lies at positive y axis")

        elif (x == len(grid)//2 and y < len(grid)/2):
            print ("lies at negative y axis")

        elif (y == len(grid)//2 and x < len(grid)/2):
            print ("lies at negative x axis")

        elif (y == len(grid)//2 and x > len(grid)/2):
            print ("lies at positive x axis")

        elif((x == len(grid)//2 and y == len(grid)/2):
            print ("lies at origin")

In reference to the second part of the question, you can simplify this greatly:

for i in range(len(grid)):
  for j in range(len(grid[i])):
    x = i - 10
    y = j + 10

This will give you the correct values of x and y in your cartesian coordinate system, so it can be compared against 0 rather than having to calculate the length every time. The rest of the code is valid but may be simpler as follows:

if x > 0:
  if y > 0:
    print("Quadrant 1")
  elif y < 0:
    print("Quadrant 4")
  else:
    print("positive x-axis")
elif x < 0:
  if y > 0:
    print("Quadrant 2")
  elif y < 0:
    print("Quadrant 3")
  else:
    print("negative x-axis")
else:
  if y == 0:
    print("Origin")
  elif y > 0:
    print("positive y-axis")
  else:
    print("negative y-axis")