给定XY平面中的三个非共线积分点，求出由这三个点形成的三角形内的积分点数。 (如果XY平面中的点的两个坐标均为积分，则称该点为积分/晶格点)。
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’s定理，该定理指出以下等式适用于简单的Polygon。
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Shoelace公式找到A (三角形区域)。
A = 1/2 * abs(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))
如何找到B(三角形边缘上的积分点数)?
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 SinghPython3
# 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_56C#
// 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输出：
Number of integral points inside given triangle is 6