Classification Of Point In Polygon

Tuesday Jul 20th 2010 by Marco Amagliani

Classify a point respect to a plane polygon that can be concave (i.e. with some angle grater than 180 degrees). Good performance, excellent stability.

ptInPolygon - Point classification

This is an algorithm to test if a point is inside, outside or on (according to an input tolerance) a closed polygon definite with an ordinate set of points.
The classical algorithm is "Ray casting algorithm" ( and counts the number of intersections between the polygon and a segment that connect the point to a point that is out for sure. It has a lot of drawbacks and special cases.
The algorithm I propose counts how many times the polygon cross from one quadrant to another. It can be considered a variant of the "Winding number algorithm", described in the same article on wikipedia, but does not uses trigonometric functions.
It is more stable than the standard one and very fast, at least as fast as the Ray casting algorithm.
All the code is in a template class:

   template < class Pt, class PtIterator = Pt* > class ptInPolygon;

that must be used in as follow:

#include "ptInPolygon.h"
Pt2d ptest;
double tol = 0.1;
ptInPolygon < Pt2d, ptsarr::iterator > xx(tol);
GeoStuff::Locate r = xx(ptest, pts.begin(), pts.end()); // operator()
if(r == GeoStuff::In) {..}
else if(r == GeoStuff::Out) {..}
else if(r == GeoStuff::On) {..}
else {/*impossible*/}

The attached example uses this algorithm to classify a grid of points with a polygon. In the image you can see red out points, green inside points and blue points found on polygon according to the input tolerance.

All the code is in ptInPolygon.h, that I copy here, if you do not want to download the example.

// ==== Start =====

// ptInPolygon.h
#pragma once

#include "math.h"

// GeoMetric Stuff (Vect, tol, and other definitions).

class GeoStuff {
GeoStuff(double tol) : m_tol2(sq(tol)){}
enum Locate{In=0, On, Out};
struct Vect {
Vect(double x=0, double y=0) { m_v[0] = x; m_v[1] = y;}
double m_v[2];
double & operator[](int i) {return m_v[i];}
double operator[](int i) const {return m_v[i];}
static inline double dot(Vect& a, Vect & b) {
return a[0]*b[0]+a[1]*b[1];
static inline double cross(Vect& a, Vect & b) {
return a[0]*b[1]-a[1]*b[0];
static inline double sq(double a) {return a*a;}

template < class Pt >
static Vect diff(const Pt& a, const Pt & b) {
return Vect(a[0]-b[0],a[1]-b[1]);
template < class Pt >
bool equiv(const Pt & a, const Pt & b){
return sq(a[0]-b[0])+sq(a[1]-b[1]) <= m_tol2;
double m_tol2;
static char quadrante(Vect & v) {
static char qq[2][2]={{0,1},{3,2}};
return qq[v[1] < 0][v[0] < 0];

// The class ptInPolygon is able to calssify points as inside, outside or on a polygon
// defined by a set of successve points.
// The point class must only have operator []
// to access x (coordinate 0) and y (coordinate 1).

template < class Pt, class PtIterator = Pt* >
class ptInPolygon : public GeoStuff {
ptInPolygon(double tol): GeoStuff(tol){}
Locate operator()(const Pt & pt, const PtIterator & begin, const PtIterator & end)
int npts = 0;
for(PtIterator it = begin; it != end; ++it)
if(equiv((*it), pt))
return On;
if(npts < 2)
return Out;
Vect vp=diff(*begin,pt);
Vect va;
int qp = quadrante(vp), qa;
int quad_crosses = 0;
for(PtIterator it = end-1; ;--it) {
va = diff((*it), pt);
qa = quadrante(va);
int diff = qa - qp;
if(diff != 0) {
double vv = dot(vp,vp), ww = dot(va,va), vw = dot(va,vp);
double den = vv+ww-vw*2;
if((vv*ww-sq(vw)) <= m_tol2 * den) {// mindist < tol
if(fabs(ww-vv) < den) // projection inside segment
return On;
switch(diff) {
case -3:
case 1:
case -1:
case 3:
case -2:
case 2:{
if(cross(vp,va) < 0)
quad_crosses -= 2;
quad_crosses += 2;
if(it == begin)
vp = va;
qp = qa;
return abs(quad_crosses) > 2 ? In : Out;

// ==== End =====

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