计算3点（x，y）的曲率
我有一个二维欧几里德空间。 给出了三点。
例如（p2是中间点）：
Point2D p1 = new Point2D.Double(177, 289); Point2D p2 = new Point2D.Double(178, 290); Point2D p3 = new Point2D.Double(178, 291); 现在我想计算这三个点的曲率 。
double curvature = calculateCurvature(p1, p2, p3);
这个怎么做？ 我有一个现有的方法（没有java外部库）？
- 曲率： https ： //en.wikipedia.org/wiki/Curvature
- Menger Curvature： https ： //en.wikipedia.org/wiki/Menger_curvature
对于Menger Curvature，公式就在维基百科文章中：
curvature = 4*triangleArea/(sideLength1*sideLength2*sideLength3)
您尝试过哪些代码？
考虑到你的3分，计算这4个值并不难。
以下是一些有用的方法：
/** * Returns twice the signed area of the triangle abc. * @param a first point * @param b second point * @param c third point * @return twice the signed area of the triangle abc */ public static double area2(Point2D a, Point2D b, Point2D c) { return (bx-ax)*(cy-ay) - (by-ay)*(cx-ax); } /** * 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); }
没有更多的事要做。 警告： area2返回一个带符号的double，具体取决于您的点的方向（顺时针或逆时针）。
从您引用的维基中 ，曲率定义为 
其中A是由三个点x，y和z（在你的情况下为p1，p2，p3）和| xy |形成的三角形所包围的区域。 是点x和y之间的距离。
将公式转换为代码，你就完成了！
正如Eric Duminil在他的回答中已经指出的那样，计算是
curvature = 4*triangleArea/(sideLength0*sideLength1*sideLength2)
我浪费了一些时间来创建这个包含computeCurvature方法的交互式示例，该方法一次完成整个计算：
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 pointList; private Point2D draggedPoint; public CurvatureFromThreePointsPanel() { this.pointList = new ArrayList (); 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 } }