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Count Integral points inside a Triangle

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Given three non-collinear integral points in XY plane, find the number of integral points inside the triangle formed by the three points. (A point in XY plane is said to be integral/lattice point if both its co-ordinates are integral).

Example: 

Input: p = (0, 0), q = (0, 5) and r = (5,0) 
Output: 6
Explanation: The points (1,1) (1,2) (1,3) (2,1) (2,2) and (3,1) are the integral points inside the triangle.

traingle


 

We can use the Pick’s theorem, which states that the following equation holds true for a simple Polygon. 

Pick's Theorem:
 A = I + (B/2) -1

A ==> Area of Polygon
B ==> Number of integral points on edges of polygon
I ==> Number of integral points inside the polygon

Using the above formula, we can deduce,
I = (2A - B + 2) / 2 

We can find A (area of triangle) using below Shoelace formula

A =  1/2 * abs(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) 


How to find B (number of integral points on edges of a triangle)? 
We can find the number of integral points between any two vertex (V1, V2) of the triangle using the following algorithm. 

1. If the edge formed by joining V1 and V2 is parallel 
   to the X-axis, then the number of integral points 
   between the vertices is : 
        abs(V1.x - V2.x) - 1

2. Similarly, if edge is parallel to the Y-axis, then 
   the number of integral points in between is :
    abs(V1.y - V2.y) - 1

3. Else, we can find the integral points between the
   vertices using below formula:
     GCD(abs(V1.x-V2.x), abs(V1.y-V2.y)) - 1
   The above formula is a well known fact and can be 
   verified using simple geometry. (Hint: Shift the 
   edge such that one of the vertex lies at the Origin.) 

Please refer below link for detailed explanation.
https://www.geeksforgeeks.org/number-integral-points-two-points/


Below is the implementation of the above algorithm. 

C++

// C++ program to find Integral points inside a triangle
#include<bits/stdc++.h>
using namespace std;
 
// Class to represent an Integral point on XY plane.
class Point
{
public:
    int x, y;
    Point(int a=0, int b=0):x(a),y(b) {}
};
 
//utility function to find GCD of two numbers
// GCD of a and b
int gcd(int a, int b)
{
    if (b == 0)
       return a;
    return gcd(b, a%b);
}
 
// Finds the no. of Integral points between
// two given points.
int getBoundaryCount(Point p,Point q)
{
    // Check if line parallel to axes
    if (p.x==q.x)
        return abs(p.y - q.y) - 1;
    if (p.y == q.y)
        return abs(p.x - q.x) - 1;
 
    return gcd(abs(p.x-q.x), abs(p.y-q.y)) - 1;
}
 
// Returns count of points inside the triangle
int getInternalCount(Point p, Point q, Point r)
{
    // 3 extra integer points for the vertices
    int BoundaryPoints = getBoundaryCount(p, q) +
                         getBoundaryCount(p, r) +
                         getBoundaryCount(q, r) + 3;
 
    // Calculate 2*A for the triangle
    int doubleArea = abs(p.x*(q.y - r.y) + q.x*(r.y - p.y)  +
                         r.x*(p.y - q.y));
 
    // Use Pick's theorem to calculate the no. of Interior points
    return (doubleArea - BoundaryPoints + 2)/2;
}
 
// driver function to check the program.
int main()
{
    Point p(0, 0);
    Point q(5, 0);
    Point r(0, 5);
    cout << "Number of integral points inside given triangle is "
         << getInternalCount(p, q, r);
    return 0;
}

                    

Java

// Java program to find Integral points inside a triangle
// Class to represent an Integral point on XY plane.
class Point
{
    int x, y;
 
    public Point(int a, int b)
    {
        x = a;
        y = b;
    }
}
 
class GFG
{
    // utility function to find GCD of two numbers
    // GCD of a and b
    static int gcd(int a, int b)
    {
        if (b == 0)
            return a;
        return gcd(b, a % b);
    }
 
    // Finds the no. of Integral points between
    // two given points.
    static int getBoundaryCount(Point p, Point q)
    {
        // Check if line parallel to axes
        if (p.x == q.x)
            return Math.abs(p.y - q.y) - 1;
 
        if (p.y == q.y)
            return Math.abs(p.x - q.x) - 1;
 
        return gcd(Math.abs(p.x - q.x),
                   Math.abs(p.y - q.y)) - 1;
    }
 
    // Returns count of points inside the triangle
    static int getInternalCount(Point p, Point q, Point r)
    {
 
        // 3 extra integer points for the vertices
        int BoundaryPoints = getBoundaryCount(p, q) +
                             getBoundaryCount(p, r) +
                             getBoundaryCount(q, r) + 3;
 
        // Calculate 2*A for the triangle
        int doubleArea = Math.abs(p.x * (q.y - r.y) +
                                  q.x * (r.y - p.y) +
                                  r.x * (p.y - q.y));
 
        // Use Pick's theorem to calculate
        // the no. of Interior points
        return (doubleArea - BoundaryPoints + 2) / 2;
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        Point p = new Point(0, 0);
        Point q = new Point(5, 0);
        Point r = new Point(0, 5);
        System.out.println("Number of integral points" +
                           " inside given triangle is " +
                              getInternalCount(p, q, r));
    }
}
 
// This code is contributed by Vivek Kumar Singh

                    

Python3

# Python3 program to find Integral
# points inside a triangle
 
# Class to represent an Integral
# point on XY plane.
class Point:
 
    def __init__(self, x, y):
        self.x = x
        self.y = y
         
# Utility function to find GCD of
# two numbers GCD of a and b
def gcd(a, b):
 
    if (b == 0):
        return a
         
    return gcd(b, a % b)
 
# Finds the no. of Integral points
# between two given points
def getBoundaryCount(p, q):
     
    # Check if line parallel to axes
    if (p.x == q.x):
        return abs(p.y - q.y) - 1
    if (p.y == q.y):
        return abs(p.x - q.x) - 1
 
    return gcd(abs(p.x - q.x),
               abs(p.y - q.y)) - 1
 
# Returns count of points inside the triangle
def getInternalCount(p, q, r):
 
    # 3 extra integer points for the vertices
    BoundaryPoints = (getBoundaryCount(p, q) +
                      getBoundaryCount(p, r) +
                      getBoundaryCount(q, r) + 3)
 
    # Calculate 2*A for the triangle
    doubleArea = abs(p.x * (q.y - r.y) +
                     q.x * (r.y - p.y) +
                     r.x * (p.y - q.y))
 
    # Use Pick's theorem to calculate
    # the no. of Interior points
    return (doubleArea - BoundaryPoints + 2) // 2
 
# Driver code
if __name__=="__main__":
     
    p = Point(0, 0)
    q = Point(5, 0)
    r = Point(0, 5)
     
    print("Number of integral points "
          "inside given triangle is ",
          getInternalCount(p, q, r))
  
# This code is contributed by rutvik_56

                    

C#

// C# program to find Integral points
// inside a triangle
using System;
 
// Class to represent an Integral point
// on XY plane.
public class Point
{
    public int x, y;
 
    public Point(int a, int b)
    {
        x = a;
        y = b;
    }
}
 
class GFG
{
    // utility function to find GCD of
    // two numbers a and b
    static int gcd(int a, int b)
    {
        if (b == 0)
            return a;
        return gcd(b, a % b);
    }
 
    // Finds the no. of Integral points between
    // two given points.
    static int getBoundaryCount(Point p, Point q)
    {
        // Check if line parallel to axes
        if (p.x == q.x)
            return Math.Abs(p.y - q.y) - 1;
 
        if (p.y == q.y)
            return Math.Abs(p.x - q.x) - 1;
 
        return gcd(Math.Abs(p.x - q.x),
                Math.Abs(p.y - q.y)) - 1;
    }
 
    // Returns count of points inside the triangle
    static int getInternalCount(Point p, Point q, Point r)
    {
 
        // 3 extra integer points for the vertices
        int BoundaryPoints = getBoundaryCount(p, q) +
                             getBoundaryCount(p, r) +
                              getBoundaryCount(q, r) + 3;
 
        // Calculate 2*A for the triangle
        int doubleArea = Math.Abs(p.x * (q.y - r.y) +
                                  q.x * (r.y - p.y) +
                                  r.x * (p.y - q.y));
 
        // Use Pick's theorem to calculate
        // the no. of Interior points
        return (doubleArea - BoundaryPoints + 2) / 2;
    }
 
    // Driver Code
    public static void Main(String[] args)
    {
        Point p = new Point(0, 0);
        Point q = new Point(5, 0);
        Point r = new Point(0, 5);
        Console.WriteLine("Number of integral points" +
                         " inside given triangle is " +
                            getInternalCount(p, q, r));
    }
}
 
// This code is contributed by 29AjayKumar

                    

Javascript

<script>
 
// JavaScript program to find Integral points inside a triangle
// Class to represent an Integral point on XY plane.
 
class Point
{
    constructor(a,b)
    {
        this.x=a;
        this.y=b;
    }
}
 
// utility function to find GCD of two numbers
    // GCD of a and b
function gcd(a,b)
{
    if (b == 0)
            return a;
        return gcd(b, a % b);
}
 
// Finds the no. of Integral points between
    // two given points.
function getBoundaryCount(p,q)
{
    // Check if line parallel to axes
        if (p.x == q.x)
            return Math.abs(p.y - q.y) - 1;
  
        if (p.y == q.y)
            return Math.abs(p.x - q.x) - 1;
  
        return gcd(Math.abs(p.x - q.x),
                   Math.abs(p.y - q.y)) - 1;
}
 
// Returns count of points inside the triangle
function getInternalCount(p,q,r)
{
    // 3 extra integer points for the vertices
        let BoundaryPoints = getBoundaryCount(p, q) +
                             getBoundaryCount(p, r) +
                             getBoundaryCount(q, r) + 3;
  
        // Calculate 2*A for the triangle
        let doubleArea = Math.abs(p.x * (q.y - r.y) +
                                  q.x * (r.y - p.y) +
                                  r.x * (p.y - q.y));
  
        // Use Pick's theorem to calculate
        // the no. of Interior points
        return (doubleArea - BoundaryPoints + 2) / 2;
}
 
// Driver Code
let p = new Point(0, 0);
let q = new Point(5, 0);
let r = new Point(0, 5);
document.write("Number of integral points" +
                           " inside given triangle is " +
                              getInternalCount(p, q, r));
 
 
 
// This code is contributed by rag2127
 
</script>

                    

Output: 

Number of integral points inside given triangle is 6

Time Complexity: O(log(min(a,b))), as we are using recursion to find the GCD.
Auxiliary Space: O(log(min(a,b))), for recursive stack space.



 



Last Updated : 30 Sep, 2022
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