Check Whether Point Is Inside The Polygon / Circle

I created a 2D convex hull and wanted to check whether a point is inside it.

If the point is inside the polygon, the neighbor cross product vectors have the same directions. If the point is outside, one vector is toward the outside, the other one is toward the inside.

Here is a point P that is closed to the vertices A, B, and C. We have

The vector is toward the outside screen, the vector is toward the inside direction. So P is not in the convex hull.

#include <iostream>
#include <vtkSmartPointer.h>
#include <vtkSphereSource.h>
#include <vtkActor.h>
#include <vtkConeSource.h>
#include <vtkRenderer.h>
#include <vtkRenderWindow.h>
#include <vtkPolyDataMapper.h>
#include <vtkTextActor.h>
#include <vtkTextProperty.h>
#include <vtkRenderWindowInteractor.h>
#include <vtkActor2D.h>
#include <vtkConvexHull2D.h>
#include <vtkDelaunay3D.h>
#include <vtkPNGReader.h>
#include <vtkImageMapper.h>
#include <vtkAppendFilter.h>
#include <vtkDataSetSurfaceFilter.h>
#include <vtkConnectedPointsFilter.h>
#include <vtkCamera.h>

#include "point.hpp"

#define vtkSPtr vtkSmartPointer
#define vtkSPtrNew(Var, Type) vtkSPtr<Type> Var = vtkSPtr<Type>::New();

using namespace std;

int main()
{
    vtkSPtrNew( points, vtkPoints );
    points->InsertNextPoint( 0, 0, 0 );
    points->InsertNextPoint( 0, 1, 0 );
    points->InsertNextPoint( 1, 0, 0 );
    points->InsertNextPoint( 1, 1, 0 );
    points->InsertNextPoint( 2, 1, 0 );
    points->InsertNextPoint( 2, 2, 0 );
    points->InsertNextPoint( 1, 2, 0 );
    vtkSPtrNew( polyData, vtkPolyData );
    polyData->SetPoints( points );

    vtkSPtrNew( hull2D, vtkConvexHull2D );
    hull2D->SetHullShape( 1 ); //0: BoundingRectangle or 1: ConvexHull.
    hull2D->SetInputData( polyData );
    hull2D->Update();

    Point pt[2] = { Point( 0.5, 0.5, 0 ), Point( -0.5, 0.5, 0 ) };
    auto data = hull2D->GetOutput();
    auto cell = data->GetCell( 0 );
    vtkIdList *cellPtIds = cell->GetPointIds();

    for( int i = 0; i < 2; ++i )
    {
        bool sameDir = true;
        for( int j = 0; j < cellPtIds->GetNumberOfIds()-1; ++j )
        {
            Point pt0( data->GetPoint( cellPtIds->GetId( j ) ) );
            Point pt1( data->GetPoint( cellPtIds->GetId( j+1 ) ) );
            Point pt2( data->GetPoint( cellPtIds->GetId( (j+2)%cellPtIds->GetNumberOfIds() ) ) );

            auto dir0 = (pt0 - pt[i])^(pt1 - pt[i]);
            auto dir1 = (pt1 - pt[i])^(pt2 - pt[i]);
            if( dir0.Dot( dir1 ) < 0 )
            {
                sameDir = false;
                break;
            }
        }
        if( !sameDir )
        {
            cout << "outside point: " << i+1 << endl;
        }
    }

    return 0;
}
/*
outside point: 2
*/