Count Integral points inside a Triangle

• Difficulty Level : Hard
• Last Updated : 11 Jan, 2022

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. Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

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.)

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#includeusing 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 bint 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 triangleint 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 bdef gcd(a, b):     if (b == 0):        return a             return gcd(b, a % b) # Finds the no. of Integral points# between two given pointsdef 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 triangledef 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 codeif __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 triangleusing 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



Output:

Number of integral points inside given triangle is 6

This article is contributed by Ashutosh Kumar. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.