 We construct the caustic curve of the cardioid relative to the cusp point O.
Click the linked diagram below and then the ‘Animate’ button to generate the caustic curve of the cardioid. Click the button again to stop the generation. Describe the mechanics of this generation. Clicking the little x at bottom right will clear the drawing from the applet window. Now try dragging the drive point Q rapidly around the circle. This is probably less useful than using the animation. What is the generated caustic curve?
What do you notice about the points O, P, Q in this construction?
It would appear that the caustic curve in this case is a nephroid, and this is in fact the case.
Why can’t we be more definite about this? Just because the curve looks like a nephroid, doesn’t mean that it is in fact a nephroid. We would have to check the equation produced in the construction. The applets are very useful for suggesting results, but not for proving them.
BIBLIOGRAPHY
A Book of Curves, Lockwood, E. H. (Cambridge University Press 1967), pages 71, 185.
|
