Dados tres puntos integrales no colineales en el plano XY, encuentre el número de puntos integrales dentro del triángulo formado por los tres puntos. (Se dice que un punto en el plano XY es integral/punto reticular si sus dos coordenadas son integrales).
Ejemplo:
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.
Podemos usar el teorema de Pick , que establece que la siguiente ecuación se cumple para un polígono simple.
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
Podemos encontrar A (área del triángulo) usando la siguiente fórmula Shoelace .
A = 1/2 * abs(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))
¿Cómo encontrar B (número de puntos integrales en los bordes de un triángulo)?
Podemos encontrar el número de puntos integrales entre dos vértices cualesquiera (V1, V2) del triángulo usando el siguiente algoritmo.
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/
A continuación se muestra la implementación del algoritmo anterior.
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>
Producción:
Number of integral points inside given triangle is 6
Complejidad de tiempo: O(log(min(a,b))), ya que estamos usando la recursividad para encontrar el GCD.
Espacio auxiliar: O(1), ya que no estamos utilizando ningún espacio adicional.
Este artículo es una contribución de Ashutosh Kumar . Si le gusta GeeksforGeeks y le gustaría contribuir, también puede escribir un artículo y enviarlo por correo a review-team@geeksforgeeks.org. Vea su artículo que aparece en la página principal de GeeksforGeeks y ayude a otros Geeks.
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Artículo escrito por GeeksforGeeks-1 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA