Point in Convex Polygon Algorithm
The Point in Convex Polygon Algorithm is a computational geometry technique used to determine if a given point lies inside, outside, or on the boundary of a convex polygon. Convex polygons are polygons where all the interior angles are less than 180 degrees and have the property that any segment connecting two points inside the polygon lies entirely within the polygon. The algorithm is useful in various applications, such as computer graphics, computer vision, and geographic information systems, where it is essential to determine the relationship between a point and a geometric shape.
The algorithm works by performing a series of binary searches on the polygon's vertices to determine the relative position of the given point. The binary search is employed to find the vertex with the smallest angle relative to the point, effectively dividing the polygon into two smaller polygons. Then, the algorithm checks which of these two polygons the point belongs to and repeats the process until it reaches a triangle. Once a triangle is formed, it is straightforward to determine if the point lies inside, outside, or on its boundary using the cross product or barycentric coordinates. The Point in Convex Polygon Algorithm is efficient, with a time complexity of O(log n) for a polygon with n vertices, making it suitable for applications that involve large datasets or real-time processing.
/*
Petar 'PetarV' Velickovic
Algorithm: Point in Convex Polygon
*/
#include <stdio.h>
#include <math.h>
#include <string.h>
#include <iostream>
#include <vector>
#include <list>
#include <string>
#include <algorithm>
#include <queue>
#include <stack>
#include <set>
#include <map>
#include <complex>
#define MAX_N 100001
using namespace std;
typedef long long lld;
struct Point
{
double X, Y;
Point()
{
this->X = 0;
this->Y = 0;
}
Point(double x, double y)
{
this->X = x;
this->Y = y;
}
};
int n;
Point Polygon[MAX_N];
//Algoritam koji odredjuje da li je tacka Q unutar datog konveksnog mnogougla
//Podrazumeva se da su temena mnogougla data u obrnotum smeru kazaljke na satu
//Moze se lako prosiriti u test preseka dva konveksna mnogougla
//Slozenost: O(log N)
inline int crossProduct(Point a, Point b, Point c)
{
return ((b.X - a.X) * (c.Y - a.Y) - (b.Y - a.Y) * (c.X - a.X));
}
inline bool isLeft(Point a, Point b, Point c)
{
return (crossProduct(a, b, c) >= 0);
}
inline bool inTriangle(Point a, Point b, Point c, Point x)
{
return (isLeft(x,a,b) == isLeft(x,b,c) && isLeft(x,b,c) == isLeft(x,c,a));
}
inline int b_search(int left, int right, Point Q)
{
int i = left;
int j = right;
while (i < j-1)
{
int mid = (i+j)/2;
if (isLeft(Polygon[0], Polygon[mid], Q)) i = mid;
else j = mid;
}
return i;
}
bool inPolygon(Point Q)
{
int poz = b_search(1, n-1, Q);
return inTriangle(Polygon[0], Polygon[poz], Polygon[poz+1], Q);
}
int main()
{
n = 5;
Polygon[0] = Point(150, 108);
Polygon[1] = Point(250, 103);
Polygon[2] = Point(289, 169);
Polygon[3] = Point(240, 237);
Polygon[4] = Point(120, 195);
printf(inPolygon(Point(179, 185)) ? "YES\n" : "NO\n");
printf(inPolygon(Point(216, 90)) ? "YES\n" : "NO\n");
printf(inPolygon(Point(129, 174)) ? "YES\n" : "NO\n");
return 0;
}
