Circle and Lattice Points

20 Jun, 2022
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
Java
Python3
C#
PHP
Javascript
// C++ program to find countLattice points on a circle#include<bits/stdc++.h>using namespace std; // Function to count Lattice points on a circleint 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 programint main(){    int r = 5;    cout << countLattice(r);    return 0;}
// Java program to find// countLattice points on a circle class GFG{ // Function to count// Lattice points on a circlestatic 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 codepublic static void main(String arg[]){    int r = 5;    System.out.println(countLattice(r));}}// This code is contributed by Anant Agarwal.
# Python3 program to find# countLattice points on a circle import math # Function to count Lattice# points on a circledef 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 programr = 5print(countLattice(r)) # This code is contributed by# Smitha Dinesh Semwal
// C# program to find countLattice// points on a circleusing 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 program to find countLattice// points on a circle // Function to count Lattice// points on a circlefunction 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?>
<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.htmlThis 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. 
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