Calculate curvature for 3 Points (x,y)

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礼貌的吻别
礼貌的吻别 2021-01-13 12:22

I have a two dimensional euclidean space. Three points are given.

For example (p2 is the middle point):

Point2D p1 = new Point2D.Double(177, 289);
Po         


        
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  • 2021-01-13 12:40

    As already pointed out by Eric Duminil in his answer, the computation is

    curvature = 4*triangleArea/(sideLength0*sideLength1*sideLength2)
    

    I wasted some time with creating this interactive example that contains a computeCurvature method that does the whole computation at once:

    import java.awt.Color;
    import java.awt.Graphics;
    import java.awt.Graphics2D;
    import java.awt.event.MouseEvent;
    import java.awt.event.MouseListener;
    import java.awt.event.MouseMotionListener;
    import java.awt.geom.Ellipse2D;
    import java.awt.geom.Point2D;
    import java.util.ArrayList;
    import java.util.List;
    
    import javax.swing.JFrame;
    import javax.swing.JPanel;
    import javax.swing.SwingUtilities;
    
    public class CurvatureFromThreePoints
    {
        public static void main(String[] args)
        {
            SwingUtilities.invokeLater(new Runnable()
            {
                @Override
                public void run()
                {
                    createAndShowGUI();
                }
            });
        }
    
        private static void createAndShowGUI()
        {
            JFrame f = new JFrame();
            f.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
            f.getContentPane().add(new CurvatureFromThreePointsPanel());
            f.setSize(800,800);
            f.setLocationRelativeTo(null);
            f.setVisible(true);
        }
    
    }
    
    class CurvatureFromThreePointsPanel extends JPanel 
        implements MouseListener, MouseMotionListener
    {
        private final List<Point2D> pointList;
        private Point2D draggedPoint;
    
        public CurvatureFromThreePointsPanel()
        {
            this.pointList = new ArrayList<Point2D>();
    
            pointList.add(new Point2D.Double(132,532));
            pointList.add(new Point2D.Double(275,258));
            pointList.add(new Point2D.Double(395,267));
    
            addMouseListener(this);
            addMouseMotionListener(this);
        }
    
        private static double computeCurvature(Point2D p0, Point2D p1, Point2D p2)
        {
            double dx1 = p1.getX() - p0.getX();
            double dy1 = p1.getY() - p0.getY();
            double dx2 = p2.getX() - p0.getX();
            double dy2 = p2.getY() - p0.getY();
            double area = dx1 * dy2 - dy1 * dx2;
            double len0 = p0.distance(p1);
            double len1 = p1.distance(p2);
            double len2 = p2.distance(p0);
            return 4 * area / (len0 * len1 * len2);
        }
    
        // Adapted from https://stackoverflow.com/a/4103418
        private static Point2D computeCircleCenter(
            Point2D p0, Point2D p1, Point2D p2)
        {
            double x0 = p0.getX();
            double y0 = p0.getY();
            double x1 = p1.getX();
            double y1 = p1.getY();
            double x2 = p2.getX();
            double y2 = p2.getY();
            double offset = x1 * x1 + y1 * y1;
            double bc = (x0 * x0 + y0 * y0 - offset) / 2.0;
            double cd = (offset - x2 * x2 - y2 * y2) / 2.0;
            double det = (x0 - x1) * (y1 - y2) - (x1 - x2) * (y0 - y1);
            double invDet = 1 / det;
            double cx = (bc * (y1 - y2) - cd * (y0 - y1)) * invDet;
            double cy = (cd * (x0 - x1) - bc * (x1 - x2)) * invDet;
            return new Point2D.Double(cx, cy);
        }
    
        @Override
        protected void paintComponent(Graphics gr)
        {
            super.paintComponent(gr);
            Graphics2D g = (Graphics2D)gr;
    
            g.setColor(Color.RED);
            for (Point2D p : pointList)
            {
                double r = 5;
                g.draw(new Ellipse2D.Double(p.getX()-r, p.getY()-r, r+r, r+r));
            }
    
            g.setColor(Color.BLACK);
            //g.draw(Paths.fromPoints(spline.getInterpolatedPoints(), false));
    
            Point2D p0 = pointList.get(0);
            Point2D p1 = pointList.get(1);
            Point2D p2 = pointList.get(2);
            double curvature = computeCurvature(p0, p1, p2);
            g.drawString("Curvature: "+curvature, 10,  20);
    
            Point2D center = computeCircleCenter(p0, p1, p2);
            double radius = center.distance(p0);
            g.draw(new Ellipse2D.Double(
                center.getX() - radius, center.getY() - radius,
                radius + radius, radius + radius));
        }
    
        @Override
        public void mouseDragged(MouseEvent e)
        {
            if (draggedPoint != null)
            {
                draggedPoint.setLocation(e.getX(), e.getY());
                repaint();
    
                System.out.println("Points: ");
                for (Point2D p : pointList)
                {
                    System.out.println("    "+p);
                }
            }
        }
    
    
        @Override
        public void mousePressed(MouseEvent e)
        {
            final double thresholdSquared = 10 * 10;
            Point2D p = e.getPoint();
            Point2D closestPoint = null;
            double minDistanceSquared = Double.MAX_VALUE;
            for (Point2D point : pointList)
            {
                double dd = point.distanceSq(p);
                if (dd < thresholdSquared && dd < minDistanceSquared)
                {
                    minDistanceSquared = dd;
                    closestPoint = point;
                }
            }
            draggedPoint = closestPoint;
        }
    
        @Override
        public void mouseReleased(MouseEvent e)
        {
            draggedPoint = null;
        }
    
        @Override
        public void mouseMoved(MouseEvent e)
        {
            // Nothing to do here
        }
    
    
        @Override
        public void mouseClicked(MouseEvent e)
        {
            // Nothing to do here
        }
    
        @Override
        public void mouseEntered(MouseEvent e)
        {
            // Nothing to do here
        }
    
    
        @Override
        public void mouseExited(MouseEvent e)
        {
            // Nothing to do here
        }
    
    
    }
    
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  • 2021-01-13 12:43

    From the wiki you referenced, the curvature is defined as

    where A is the area enclosed by the triangle formed by the three points, x, y and z (p1, p2, p3 in your case) and |x-y| is the distance between points x and y.

    Translate the formula to code and you're done!

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  • 2021-01-13 12:47

    C/C++

    // https://www.mathopenref.com/coordtrianglearea.html
    float getAreaOfTriangle(Point2f A, Point2f B, Point2f C)
    {
        return fabs(
                (A.x * (B.y - C.y) + B.x * (C.y - A.y) + C.x * (A.y - B.y)) / 2);
    }
    
    float getDistFromPtToPt(Point2f pt1, Point2f pt2)
    {
        return sqrt((pt2.x - pt1.x) * (pt2.x - pt1.x) +
                    (pt2.y - pt1.y) * (pt2.y - pt1.y));
    }
    
    
    // https://en.wikipedia.org/wiki/Menger_curvature
    float
    getCurvatureUsingTriangle(Point2f pt1, Point2f pt2, Point2f pt3, bool bDebug)
    {
        float fAreaOfTriangle = getAreaOfTriangle(pt1, pt2, pt3);
        float fDist12 = getDistFromPtToPt(pt1, pt2);
        float fDist23 = getDistFromPtToPt(pt2, pt3);
        float fDist13 = getDistFromPtToPt(pt1, pt3);
        float fKappa = 4 * fAreaOfTriangle / (fDist12 * fDist23 * fDist13);
        return fKappa;
    }
    
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  • 2021-01-13 13:02

    For the Menger Curvature, the formula is right there in the Wikipedia article :

    curvature = 4*triangleArea/(sideLength1*sideLength2*sideLength3)

    Which code did you try exactly?

    It shouldn't be too hard to calculate those 4 values given your 3 points.

    Here are some helpful methods :

    /**
     * Returns twice the signed area of the triangle a-b-c.
     * @param a first point
     * @param b second point
     * @param c third point
     * @return twice the signed area of the triangle a-b-c
     */
    public static double area2(Point2D a, Point2D b, Point2D c) {
        return (b.x-a.x)*(c.y-a.y) - (b.y-a.y)*(c.x-a.x);
    }
    
    /**
     * Returns the Euclidean distance between this point and that point.
     * @param that the other point
     * @return the Euclidean distance between this point and that point
     */
    public double distanceTo(Point2D that) {
        double dx = this.x - that.x;
        double dy = this.y - that.y;
        return Math.sqrt(dx*dx + dy*dy);
    }
    

    There's not much more to do. Warning : area2 returns a signed double, depending on the orientation of your points (clockwise or anticlockwise).

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