# How to check if a point (x,y) is inside a polygon in the Cartesian coordinate system?

This question already has an answer here: Point in Polygon aka hit test C# Point in polygon

Given a random polygon formulated with N line equations in the Cartesian coordinate system, is there any standard formula that is used to check for membership of a point (x,y)?

The simple solution is to get all the line formulas and check if point X is below this line, above that line and to the right of the other line, etc. But this will probably be tedious.

I should note that the polygon can be of any shape with any number of sides and may concave or convex.

```float slope(CGPoint p1, CGPoint p2)
{
return (p2.y - p1.y) / (p2.x - p1.x);
}

CGPoint pointOnLineWithY(CGPoint p, float m, float y)
{
float x = (y - p.y)/m + p.x;
return CGPointMake(x,y);
}

CGPoint pointOnLineWithX(CGPoint p, float m, float x)
{
float y = m*(x - p.x) + p.y;
return CGPointMake(x, y);
}
```

If you have the vertices, you can compute the sum of the angles made between the test point and each pair of points making up the polygon. If it is 2*pi, then it is an interior point. If it is 0, then it is an exterior point.

Some code:

```    typedef struct {
int h,v;
} Point;

int InsidePolygon(Point *polygon,int n,Point p)
{
int i;
double angle=0;
Point p1,p2;

for (i=0;i<n;i++) {
p1.h = polygon[i].h - p.h;
p1.v = polygon[i].v - p.v;
p2.h = polygon[(i+1)%n].h - p.h;
p2.v = polygon[(i+1)%n].v - p.v;
angle += Angle2D(p1.h,p1.v,p2.h,p2.v);
}

if (ABS(angle) < PI)
return(FALSE);
else
return(TRUE);
}

/*
Return the angle between two vectors on a plane
The angle is from vector 1 to vector 2, positive anticlockwise
The result is between -pi -> pi
*/
double Angle2D(double x1, double y1, double x2, double y2)
{
double dtheta,theta1,theta2;

theta1 = atan2(y1,x1);
theta2 = atan2(y2,x2);
dtheta = theta2 - theta1;
while (dtheta > PI)
dtheta -= TWOPI;
while (dtheta < -PI)
dtheta += TWOPI;

return(dtheta);
}
```

Other places you can take a look at: http://alienryderflex.com/polygon/

http://www.ecse.rpi.edu/Homepages/wrf/Research/Short_Notes/pnpoly.html

http://sidvind.com/wiki/Point-in-polygon:_Jordan_Curve_Theorem