计算三角形内的积分点

给定XY平面内三个不共线的积分点，求这三个点构成的三角形内积分点的个数。 （如果 XY 平面中的点的两个坐标都是积分，则称其为积分/点阵点）。

例子：

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.

我们可以使用 Pick 定理，该定理指出以下等式适用于简单的多边形。

Pick's Theeorem:
 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 

我们可以使用下面的鞋带公式找到A （三角形的面积）。

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

如何找到 B（三角形边上的积分点数）？
我们可以使用以下算法找到三角形的任意两个顶点 (V1, V2) 之间的积分点数。

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/

下面是上述算法的实现。

C++

// C++ program to find Integral points inside a triangle
#include
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

 
// 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
 

输出：

Number of integral points inside given triangle is 6

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