How to create a curve between two points in 2D and return points that make this curve at each distance d?

Question

I am not good at math.

I have 2 points, A(x1, y1) and B(x2, y2) in 2D.

I need to create a virtual path from point A to B , curved in R (radius), and then return an array of points describing this curved path, not all, maybe each D (distance) from each other.

In Java, I need a method like this:

 private ArrayList<PointF> generateCurve(PointF pFrom,PointF pTo,float pRadius,float pMinDistance){ ArrayList<PointF> pOutPut = new ArrayList<PointF>(); // ...generate result to pOutPut return pOutPut; } 

How to do it?

Answer 1

I did not give up, and I worked on it for several more hours. And here is the result:

I created a method in which you can specify whether you want to have the shortest of the longest arc between points.

Here are a few calls to it, with the release:

 generateCurve(pFrom, pTo, 100f, 7f, false, false); 

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 generateCurve(pFrom, pTo, 100f, 7f, true, false); 

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 generateCurve(pFrom, pTo, 100f, 7f, false, true); 

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 generateCurve(pFrom, pTo, 100f, 7f, true, true); 

As you can see, it works like a charm. Here is the code:

 package curve; import java.awt.BasicStroke; import java.awt.Color; import java.awt.Graphics2D; import java.awt.RenderingHints; import java.awt.geom.Ellipse2D; import java.awt.geom.Line2D; import java.awt.geom.Rectangle2D; import java.awt.image.BufferedImage; import java.io.File; import java.io.IOException; import java.util.ArrayList; import java.util.List; import javax.imageio.ImageIO; /** * * @author martijn */ public class Main { /** * @param args the command line arguments */ public static void main(String[] args) throws IOException { PointF pFrom = new PointF(-10f, 30.0f); PointF pTo = new PointF(-100f, 0.0f); List<PointF> points = generateCurve(pFrom, pTo, 100f, 7f, true, true); System.out.println(points); // Calculate the bounds of the curve Rectangle2D.Float bounds = new Rectangle2D.Float(points.get(0).x, points.get(0).y, 0, 0); for (int i = 1; i < points.size(); ++i) { bounds.add(points.get(i).x, points.get(i).y); } bounds.add(pFrom.x, pFrom.y); bounds.add(pTo.x, pTo.y); BufferedImage img = new BufferedImage((int) (bounds.width - bounds.x + 50), (int) (bounds.height - bounds.y + 50), BufferedImage.TYPE_4BYTE_ABGR_PRE); Graphics2D g = img.createGraphics(); g.setRenderingHint(RenderingHints.KEY_ANTIALIASING, RenderingHints.VALUE_ANTIALIAS_ON); g.translate(25.0f - bounds.getX(), 25.0f - bounds.getY()); g.setStroke(new BasicStroke(1.0f)); g.setColor(Color.DARK_GRAY); g.drawLine(-1000, 0, 1000, 0); g.drawLine(0, -1000, 0, 1000); g.setColor(Color.RED); for (int i = 0; i < points.size(); ++i) { if (i > 0) { Line2D.Float f = new Line2D.Float(points.get(i - 1).x, points.get(i - 1).y, points.get(i).x, points.get(i).y); System.out.println("Dist : " + f.getP1().distance(f.getP2())); // g.draw(f); } g.fill(new Ellipse2D.Float(points.get(i).x - 0.8f, points.get(i).y - 0.8f, 1.6f, 1.6f)); } g.setColor(Color.BLUE); g.fill(new Ellipse2D.Float(pFrom.x - 1, pFrom.y - 1, 3, 3)); g.fill(new Ellipse2D.Float(pTo.x - 1, pTo.y - 1, 3, 3)); g.dispose(); ImageIO.write(img, "PNG", new File("result.png")); } static class PointF { public float x, y; public PointF(float x, float y) { this.x = x; this.y = y; } @Override public String toString() { return "(" + x + "," + y + ")"; } } private static List<PointF> generateCurve(PointF pFrom, PointF pTo, float pRadius, float pMinDistance, boolean shortest, boolean side) { List<PointF> pOutPut = new ArrayList<PointF>(); // Calculate the middle of the two given points. PointF mPoint = new PointF(pFrom.x + pTo.x, pFrom.y + pTo.y); mPoint.x /= 2.0f; mPoint.y /= 2.0f; System.out.println("Middle Between From and To = " + mPoint); // Calculate the distance between the two points float xDiff = pTo.x - pFrom.x; float yDiff = pTo.y - pFrom.y; float distance = (float) Math.sqrt(xDiff * xDiff + yDiff * yDiff); System.out.println("Distance between From and To = " + distance); if (pRadius * 2.0f < distance) { throw new IllegalArgumentException("The radius is too small! The given points wont fall on the circle."); } // Calculate the middle of the expected curve. float factor = (float) Math.sqrt((pRadius * pRadius) / ((pTo.x - pFrom.x) * (pTo.x - pFrom.x) + (pTo.y - pFrom.y) * (pTo.y - pFrom.y)) - 0.25f); PointF circleMiddlePoint = new PointF(0, 0); if (side) { circleMiddlePoint.x = 0.5f * (pFrom.x + pTo.x) + factor * (pTo.y - pFrom.y); circleMiddlePoint.y = 0.5f * (pFrom.y + pTo.y) + factor * (pFrom.x - pTo.x); } else { circleMiddlePoint.x = 0.5f * (pFrom.x + pTo.x) - factor * (pTo.y - pFrom.y); circleMiddlePoint.y = 0.5f * (pFrom.y + pTo.y) - factor * (pFrom.x - pTo.x); } System.out.println("Middle = " + circleMiddlePoint); // Calculate the two reference angles float angle1 = (float) Math.atan2(pFrom.y - circleMiddlePoint.y, pFrom.x - circleMiddlePoint.x); float angle2 = (float) Math.atan2(pTo.y - circleMiddlePoint.y, pTo.x - circleMiddlePoint.x); // Calculate the step. float step = pMinDistance / pRadius; System.out.println("Step = " + step); // Swap them if needed if (angle1 > angle2) { float temp = angle1; angle1 = angle2; angle2 = temp; } boolean flipped = false; if (!shortest) { if (angle2 - angle1 < Math.PI) { float temp = angle1; angle1 = angle2; angle2 = temp; angle2 += Math.PI * 2.0f; flipped = true; } } for (float f = angle1; f < angle2; f += step) { PointF p = new PointF((float) Math.cos(f) * pRadius + circleMiddlePoint.x, (float) Math.sin(f) * pRadius + circleMiddlePoint.y); pOutPut.add(p); } if (flipped ^ side) { pOutPut.add(pFrom); } else { pOutPut.add(pTo); } return pOutPut; } } 

Enjoy it!
PS: I created two math questions to solve your question:

• Analytical geometry: coordinates of a point, same distance from two points.
• Trigonometry: Solve (1-cosα) 2 + sin2α = d2 for α

Answer 2

It works:

 private static double GetAngle(Point2D x, Point2D o, double R){ double cosa = (x.getX()-o.getX())/R; double sina = (x.getY()-o.getY())/R; double angle = Math.acos(cosa); return Math.sin(angle)*sina >= 0 ? angle : 2*Math.PI - angle; } private static ArrayList<Point2D> generateCurve(Point2D pFrom,Point2D pTo,float pRadius,float pMinDistance){ ArrayList<Point2D> pOutPut = new ArrayList<Point2D>(); double dist = pFrom.distance(pTo); double h = Math.sqrt(pRadius * pRadius - (dist * dist / 4.0)); double angleStep = pMinDistance/pRadius; if(2*pRadius <= dist) throw new Error("Radius is too small"); //find center double x1 = pFrom.getX(), x2 = pFrom.getY(); double y1 = pTo.getX(), y2 = pTo.getY(); double m1 = (x1+y1)/2, m2 = (x2+y2)/2; double u1 = - (y2-x2)/dist, u2 = (y1-x1)/dist; double o1 = m1 + h * u1, o2 = m2 + h * u2; Point2D o = new Point2D.Double(o1, o2); double startAngle = GetAngle(pFrom, o, pRadius); double endAngle = GetAngle(pTo, o, pRadius); if(endAngle < startAngle) endAngle += 2 * Math.PI; for(double a = startAngle; a < endAngle; a+=angleStep){ pOutPut.add(new Point2D.Double(o1+pRadius*Math.cos(a), o2+pRadius*Math.sin(a))); } pOutPut.add(pTo); return pOutPut; } 

Here is what I get when I call it this way: generateCurve(new Point2D.Double(10,10), new Point2D.Double(400, 400), 300, 15)