Render only the segment/area of a circle that intersects the main circle

I a开发者_开发百科bsolutely love maths (or 'math' as most of you would say!) but I haven't done it to a level where I know the answer to this problem. I have a main circle which could have a centre point at any x and y on a display. Other circles will move around the display at will but at any given call to a render method I want to render not only those circles that intersect the main circle, but also only render the segment of that circle that is visible inside the main circle. An analogy would be a shadow cast on a real life object, and I only want to draw the part of that object that is 'illuminated'.

I want to do this preferably in Java, but if you have a raw formula that would be appreciated. I wonder how one might draw the shape and fill it in Java, I'm sure there must be some variation on a polyline with arcs or something?

Many thanks

Let A and B be the 2 intersection points (you can ignore it when there is no, or 1 intercetion point).

Then calculate the length of the circular line segment between A and B.

With this information, you should be able to draw the arc using Graphics' drawArc(...) method (if I'm not mistaken...).

EDIT

Well, you don't even need the length of the circular line segment. I had the line-intersection code laying around, so I built a small GUI around it how you could paint/view the ARC of such intersecting circles (there are a bit of comments in the code):

import javax.swing.*;
import java.awt.*;
import java.awt.event.*;
import java.awt.geom.Arc2D;

/**
 * @author: Bart Kiers
 */
public class GUI extends JFrame {

    private GUI() {
        super("Circle Intersection Demo");
        initGUI();
    }

    private void initGUI() {
        super.setSize(600, 640);
        super.setDefaultCloseOperation(EXIT_ON_CLOSE);
        super.setLayout(new BorderLayout(5, 5));

        final Grid grid = new Grid();

        grid.addMouseMotionListener(new MouseMotionAdapter() {
            @Override
            public void mouseDragged(MouseEvent e) {
                Point p = new Point(e.getX(), e.getY()).toCartesianPoint(grid.getWidth(), grid.getHeight());
                grid.showDraggedCircle(p);
            }
        });

        grid.addMouseListener(new MouseAdapter() {
            @Override
            public void mouseReleased(MouseEvent e) {
                Point p = new Point(e.getX(), e.getY()).toCartesianPoint(grid.getWidth(), grid.getHeight());
                grid.released(p);
            }

            @Override
            public void mousePressed(MouseEvent e) {
                Point p = new Point(e.getX(), e.getY()).toCartesianPoint(grid.getWidth(), grid.getHeight());
                grid.pressed(p);
            }
        });

        super.add(grid, BorderLayout.CENTER);
        super.setVisible(true);
    }

    public static void main(String[] args) {
        SwingUtilities.invokeLater(new Runnable() {
            @Override
            public void run() {
                new GUI();
            }
        });
    }

    private static class Grid extends JPanel {

        private Circle c1 = null;
        private Circle c2 = null;
        private Point screenClick = null;
        private Point currentPosition = null;

        public void released(Point p) {
            if (c1 == null || c2 != null) {
                c1 = new Circle(screenClick, screenClick.distance(p));
                c2 = null;
            } else {
                c2 = new Circle(screenClick, screenClick.distance(p));
            }
            screenClick = null;
            repaint();
        }

        public void pressed(Point p) {
            if(c1 != null && c2 != null) {
                c1 = null;
                c2 = null;
            }
            screenClick = p;
            repaint();
        }

        @Override
        public void paintComponent(Graphics g) {

            Graphics2D g2d = (Graphics2D) g;
            g2d.setRenderingHint(RenderingHints.KEY_ANTIALIASING, RenderingHints.VALUE_ANTIALIAS_ON);

            g2d.setColor(Color.WHITE);
            g2d.fillRect(0, 0, super.getWidth(), super.getHeight());

            final int W = super.getWidth();
            final int H = super.getHeight();
            g2d.setColor(Color.LIGHT_GRAY);
            g2d.drawLine(0, H / 2, W, H / 2); // x-axis
            g2d.drawLine(W / 2, 0, W / 2, H); // y-axis

            if (c1 != null) {
                g2d.setColor(Color.RED);
                c1.drawOn(g2d, W, H);
            }

            if (c2 != null) {
                g2d.setColor(Color.ORANGE);
                c2.drawOn(g2d, W, H);
            }

            if (screenClick != null && currentPosition != null) {
                g2d.setColor(Color.DARK_GRAY);
                g2d.setComposite(AlphaComposite.getInstance(AlphaComposite.SRC_OVER, 0.5f));
                Circle temp = new Circle(screenClick, screenClick.distance(currentPosition));
                temp.drawOn(g2d, W, H);
                currentPosition = null;
            }

            if (c1 != null && c2 != null) {

                g2d.setColor(Color.BLUE);
                g2d.setComposite(AlphaComposite.getInstance(AlphaComposite.SRC_OVER, 0.4f));
                Point[] ips = c1.intersections(c2);
                for (Point ip : ips) {
                    ip.drawOn(g, W, H);
                }
                g2d.setComposite(AlphaComposite.getInstance(AlphaComposite.SRC_OVER, 0.2f));
                if (ips.length == 2) {
                    g2d.setStroke(new BasicStroke(10.0f));
                    c1.highlightArc(g2d, ips[0], ips[1], W, H);
                }
            }

            g2d.dispose();
        }

        public void showDraggedCircle(Point p) {
            currentPosition = p;
            repaint();
        }
    }

    private static class Circle {

        public final Point center;
        public final double radius;

        public Circle(Point center, double radius) {
            this.center = center;
            this.radius = radius;
        }

        public void drawOn(Graphics g, int width, int height) {
            // translate Cartesian(x,y) to Screen(x,y)
            Point screenP = center.toScreenPoint(width, height);
            int r = (int) Math.rint(radius);
            g.drawOval((int) screenP.x - r, (int) screenP.y - r, r + r, r + r);

            // draw the center
            Point screenCenter = center.toScreenPoint(width, height);
            r = 4;
            g.drawOval((int) screenCenter.x - r, (int) screenCenter.y - r, r + r, r + r);
        }

        public void highlightArc(Graphics2D g2d, Point p1, Point p2, int width, int height) {

            double a = center.degrees(p1);
            double b = center.degrees(p2);

            // translate Cartesian(x,y) to Screen(x,y)
            Point screenP = center.toScreenPoint(width, height);
            int r = (int) Math.rint(radius);

            // find the point to start drawing our arc
            double start = Math.abs(a - b) < 180 ? Math.min(a, b) : Math.max(a, b);

            // find the minimum angle to go from `start`-angle to the other angle
            double extent = Math.abs(a - b) < 180 ? Math.abs(a - b) : 360 - Math.abs(a - b);

            // draw the arc
            g2d.draw(new Arc2D.Double((int) screenP.x - r, (int) screenP.y - r, r + r, r + r, start, extent, Arc2D.OPEN));
        }

        public Point[] intersections(Circle that) {

            // see: http://mathworld.wolfram.com/Circle-CircleIntersection.html
            double d = this.center.distance(that.center);
            double d1 = ((this.radius * this.radius) - (that.radius * that.radius) + (d * d)) / (2 * d);
            double h = Math.sqrt((this.radius * this.radius) - (d1 * d1));
            double x3 = this.center.x + (d1 * (that.center.x - this.center.x)) / d;
            double y3 = this.center.y + (d1 * (that.center.y - this.center.y)) / d;
            double x4_i = x3 + (h * (that.center.y - this.center.y)) / d;
            double y4_i = y3 - (h * (that.center.x - this.center.x)) / d;
            double x4_ii = x3 - (h * (that.center.y - this.center.y)) / d;
            double y4_ii = y3 + (h * (that.center.x - this.center.x)) / d;

            if (Double.isNaN(x4_i)) {
                // no intersections
                return new Point[0];
            }

            // create the intersection points
            Point i1 = new Point(x4_i, y4_i);
            Point i2 = new Point(x4_ii, y4_ii);

            if (i1.distance(i2) < 0.0000000001) {
                // i1 and i2 are (more or less) the same: a single intersection
                return new Point[]{i1};
            }

            // two unique intersections
            return new Point[]{i1, i2};
        }

        @Override
        public String toString() {
            return String.format("{center=%s, radius=%.2f}", center, radius);
        }
    }

    private static class Point {

        public final double x;
        public final double y;

        public Point(double x, double y) {
            this.x = x;
            this.y = y;
        }

        public double degrees(Point that) {
            double deg = Math.toDegrees(Math.atan2(that.y - this.y, that.x - this.x));
            return deg < 0.0 ? deg + 360 : deg;
        }

        public double distance(Point that) {
            double dX = this.x - that.x;
            double dY = this.y - that.y;
            return Math.sqrt(dX * dX + dY * dY);
        }

        public void drawOn(Graphics g, int width, int height) {
            // translate Cartesian(x,y) to Screen(x,y)
            Point screenP = toScreenPoint(width, height);
            int r = 7;
            g.fillOval((int) screenP.x - r, (int) screenP.y - r, r + r, r + r);
        }

        public Point toCartesianPoint(int width, int height) {
            double xCart = x - (width / 2);
            double yCart = -(y - (height / 2));
            return new Point(xCart, yCart);
        }

        public Point toScreenPoint(int width, int height) {
            double screenX = x + (width / 2);
            double screenY = -(y - (height / 2));
            return new Point(screenX, screenY);
        }

        @Override
        public String toString() {
            return String.format("(%.2f,%.2f)", x, y);
        }
    }
}

If you start the GUI above and then type 100 0 130 -80 55 180 in the text box and hit return, you'll see the following: ...

Changed the code so that circles can be drawn by pressing- and dragging the mouse. Screenshot:

Assuming you know the center point and the radius of the two circles:

1. Calculate the points where the circles intersect. This can easily be done with trigonometry. There may be no intersection (distance between the center points is longer than the sum of the radiuses, ignorable in your case), one point (distance between center points is equal to the sum of the radiuses, ignorable), or two. Special cases: the circles are identical, or the moving circle ist smaller and completely inside the main circle.

2. If there are two intersection points: take the center point from the moving circle and draw an arc between those points.

(I have no code for you, but since you love maths... ;-)