Whats an easy way to fill a Concave PathGeometry to be Convex (finding the concave vertices & removing them)?

Question

I've got a PathGeometry (polygon) built up of LineSegments on one PathFigure and I'd like to ensure that it's Convex. I have a method using the CrossProduct to determine whether the geometry is Convex, I was assuming that I could just return back a list of points that make it concave when it's false and remove those points to fill the polygon but it's not working quite right.

Here's the code I've got:

    public static bool IsConvexPolygon(this IList<Point> polygon, out List<Point> concavePoints)
    {
        int n = polygon.Count;
        List<double> result = new List<double>();
        concavePoints = new List<Point>();
        for (int i = 0; i < n; i++)
        {
            result.Add(polygon[i].CrossProduct(polygon[i.RotateNext(n)]));
            if (result.Last() < 0.0)
            {
                concavePoints.Add(polygon[i.RotateNext(n)]);
            }
        }
        return (result.All(d => d >= 0.0));
    }

    public static double CrossProduct(this Point p1, Point p2)
        {
            return (p1.X * p2.Y) - (p1.Y * p2.X);
        }

    public static int RotateNext(this int index, int count)
        {
            return (index + 1) % count;
        }

    public static PointCollection ExtractPoints(this Geometry geometry)
        {
            PointCollection pc = new PointCollection();
            if (geometry is LineGeometry)
            {
                var lg = (LineGeometry)geometry;
                pc.Add(lg.StartPoint);
                pc.Add(lg.EndPoint);
                return pc;
            }
            else if (geometry is PathGeometry)
            {
                var pg = (PathGeometry)geometry;
                if (pg.Figures.Count > 0)
                {
                    List<Point> points;
                    if ((pg.Figures[0].Segments.Count > 0) && (pg.Figures[0].Segments[0] is PolyLineSegment))
                        points = ((PolyLineSegment)pg.Figures[0].Segments[0]).Points.ToList();
                    else
                        points = pg.Figures[0].Segments.Select(seg => (seg as LineSegment).Point).ToList();

                    pc.Add(pg.Figures[0].StartPoint);
                    foreach (Point p in points)
                        pc.Add(p);
                    return pc;
                }
            }
            else if (geometry is RectangleGeometry)
            {
                var rg = (RectangleGeometry)geometry;
                var rect = rg.Rect;
                pc.Add(rect.TopLeft);
                pc.Add(rect.TopRight);
                pc.Add(rect.BottomRight);
                pc.Add(rect.BottomLeft);
                return pc;
            }
            return pc;
        }

public static Geometry CreateGeometryFromPoints(this List<Point> pts)
{
    if (pts.Count < 2)
        return null;

    PathFigure pFig = new PathFigure() { StartPoint = pts[0] };
    for (int i = 1; i < pts.Count; i++)
    {
        pFig.Segments.Add(new LineSegment(pts[i], true));
    }
    pFig.IsClosed = true;

    PathGeometry pg = new PathGeometry(new List<PathFigure>() { pFig });
    return pg;
}
public static Path CreatePolygonFromGeometry(this Geometry geo, Brush fillBrush)
        {
            Path path = new Path() { Stroke = Brushes.Black, StrokeThickness = 1, Fill = fillBrush };
            path.Data = geo;
            return path;
        }

And Here's where I making the check and correcting the polygon:

        List<Point> outstuff;
        if (geo1.ExtractPoints().IsConvexPolygon(out outstuff) == false)
        {
            // Got to fill it in if it's concave
            var newpts = geo1.ExtractPoints().Except(outstuff).ToList();
            var z = newpts.CreateGeometryFromPoints().CreatePolygonFromGeometry(Brushes.Purple);
            z.MouseRightButtonDown += delegate { canvas.Children.Remove(z); };
            canvas.Children.Add(z);
        }

Ultimately I'd like to be able to make my Concave Geometry into a Convex one like this:

Answer 1

I'd compute the convex hull (also: NTS) and remove any vertexes on the interior of the resulting convex hull polygon (using a point-in-polygon test).