Count Integral points inside a Triangle
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.
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.w3wiki.net/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.
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