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LIMAÇON : Caustic curves
Click the diagram below to open the applet, and then click the ‘Animate’ button to generate the caustic curve of the limaçon with respect to the radiant point M. Clicking the little red x at bottom right will clear the drawing from the applet window. Now drag the red point Q slowly around the circle. Can you describe the construction? Moving the point K along the horizontal line will change the shape of the limaçon, and hence the caustic curve. You can also choose different positions along the x-axis for the point M. Use the little red x to clear the screen after each such move. Are any of the reulting pedal curves familiar?
When K lies at C the limacon becomes a cardioid. If M is placed at the cusp of this curve, the caustic becomes a nephroid curve. Varying the positions of M and K gives rise to a various distortions and adaptations of the nephroid. |
Because there is a whole family of limaçons, as well as a whole family of radiant points, the problem of finding the caustic curves becomes much more interesting, if not particularly rewarding!
