Circle and Lattice Points

Given a circle of radius r in 2-D with origin or (0, 0) as center. The task is to find the total lattice points on circumference. Lattice Points are points with coordinates as integers in 2-D space.
Example: 

Input  : r = 5.
Output : 12
Below are lattice points on a circle with
radius 5 and origin as (0, 0).
(0,5), (0,-5), (5,0), (-5,0),
(3,4), (-3,4), (-3,-4), (3,-4),
(4,3), (-4,3), (-4,-3), (4,-3).
are 12 lattice point.

To find lattice points, we basically need to find values of (x, y) which satisfy the equation x² + y² = r². 
For any value of (x, y) that satisfies the above equation we actually have total 4 different combination which that satisfy the equation. For example if r = 5 and (3, 4) is a pair which satisfies the equation, there are actually 4 combinations (3, 4) , (-3,4) , (-3,-4) , (3,-4). There is an exception though, in case of (0, r) or (r, 0) there are actually 2 points as there is no negative 0.

// Initialize result as 4 for (r, 0), (-r. 0),
// (0, r) and (0, -r)
result = 4

Loop for x = 1 to r-1 and do following for every x.
    If r*r - x*x is a perfect square, then add 4 
    tor result.  

Below is the implementation of above idea. 

CPP

// C++ program to find countLattice points on a circle
#include<bits/stdc++.h>
using namespace std;

// Function to count Lattice points on a circle
int countLattice(int r)
{
    if (r <= 0)
        return 0;

    // Initialize result as 4 for (r, 0), (-r. 0),
    // (0, r) and (0, -r)
    int result = 4;

    // Check every value that can be potential x
    for (int x=1; x<r; x++)
    {
        // Find a potential y
        int ySquare = r*r - x*x;
        int y = sqrt(ySquare);

        // checking whether square root is an integer
        // or not. Count increments by 4 for four
        // different quadrant values
        if (y*y == ySquare)
            result += 4;
    }

    return result;
}

// Driver program
int main()
{
    int r = 5;
    cout << countLattice(r);
    return 0;
}

Java

// Java program to find
// countLattice points on a circle

class GFG
{

// Function to count
// Lattice points on a circle
static int countLattice(int r)
{
    if (r <= 0)
        return 0;

    // Initialize result as 4 for (r, 0), (-r. 0),
    // (0, r) and (0, -r)
    int result = 4;

    // Check every value that can be potential x
    for (int x=1; x<r; x++)
    {
        // Find a potential y
        int ySquare = r*r - x*x;
        int y = (int)Math.sqrt(ySquare);

        // checking whether square root is an integer
        // or not. Count increments by 4 for four
        // different quadrant values
        if (y*y == ySquare)
            result += 4;
    }

    return result;
}

// Driver code
public static void main(String arg[])
{
    int r = 5;
    System.out.println(countLattice(r));
}
}
// This code is contributed by Anant Agarwal.

Python3

# Python3 program to find
# countLattice points on a circle

import math

# Function to count Lattice
# points on a circle
def countLattice(r):

    if (r <= 0):
        return 0

    # Initialize result as 4 for (r, 0), (-r. 0),
    # (0, r) and (0, -r)
    result = 4

    # Check every value that can be potential x
    for x in range(1, r):

        # Find a potential y
        ySquare = r*r - x*x
        y = int(math.sqrt(ySquare))

        # checking whether square root is an integer
        # or not. Count increments by 4 for four
        # different quadrant values
        if (y*y == ySquare):
            result += 4


    return result


# Driver program
r = 5
print(countLattice(r))

# This code is contributed by
# Smitha Dinesh Semwal

C#

// C# program to find countLattice
// points on a circle
using System;

class GFG {

    // Function to count Lattice
    // points on a circle
    static int countLattice(int r)
    {
        if (r <= 0)
            return 0;

        // Initialize result as 4
        // for (r, 0), (-r. 0),
        // (0, r) and (0, -r)
        int result = 4;

        // Check every value that
        // can be potential x
        for (int x = 1; x < r; x++)
        {

            // Find a potential y
            int ySquare = r*r - x*x;
            int y = (int)Math.Sqrt(ySquare);

            // checking whether square root
            // is an integer or not. Count
            // increments by 4 for four
            // different quadrant values
            if (y*y == ySquare)
                result += 4;
        }

        return result;
    }

    // Driver code
    public static void Main()
    {
        int r = 5;

        Console.Write(countLattice(r));
    }
}

// This code is contributed by nitin mittal.

PHP

<?php
// PHP program to find countLattice
// points on a circle

// Function to count Lattice
// points on a circle
function countLattice($r)
{
    if ($r <= 0)
        return 0;

    // Initialize result as 4
    // for (r, 0), (-r. 0),
    // (0, r) and (0, -r)
    $result = 4;

    // Check every value that
    // can be potential x
    for ($x = 1; $x < $r; $x++)
    {

        // Find a potential y
        $ySquare = $r * $r - $x * $x;
        $y = ceil(sqrt($ySquare));

        // checking whether square
        // root is an integer
        // or not. Count increments
        // by 4 for four different
        // quadrant values
        if ($y * $y == $ySquare)
            $result += 4;
    }

    return $result;
}

    // Driver Code
    $r = 5;
    echo countLattice($r);

// This code is contributed by nitin mittal
?>

Javascript

<script>
// javascript program to find
// countLattice points on a circle

    // Function to count
    // Lattice points on a circle
    function countLattice(r) {
        if (r <= 0)
            return 0;

        // Initialize result as 4 for (r, 0), (-r. 0),
        // (0, r) and (0, -r)
        var result = 4;

        // Check every value that can be potential x
        for (x = 1; x < r; x++)
        {

            // Find a potential y
            var ySquare = r * r - x * x;
            var y = parseInt( Math.sqrt(ySquare));

            // checking whether square root is an integer
            // or not. Count increments by 4 for four
            // different quadrant values
            if (y * y == ySquare)
                result += 4;
        }

        return result;
    }

    // Driver code
        var r = 5;
        document.write(countLattice(r));

// This code is contributed by umadevi9616
</script>

Output: 

12

Time Complexity: O(1)
Auxiliary Space: O(1)

Reference:
http://mathworld.wolfram.com/CircleLatticePoints.html
This article is contributed by Shivam Pradhan (anuj_charm). If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.