Lame Curves

Cartesian equation:
(x/a)n + (y/b)n = 1

Click below to see one of the Associated curves.

Definitions of the Associated curves Evolute
Involute 1 Involute 2
Inverse curve wrt origin Inverse wrt another circle
Pedal curve wrt origin Pedal wrt another point
Negative pedal curve wrt origin Negative pedal wrt another point
Caustic wrt horizontal rays Caustic curve wrt another point

Click THIS LINK to experiment interactively with this curve and its associated curves.

In 1818 Lamé discussed the curves with equation given above. He considered more general curves than just those where n is an integer. If n is a rational then the curve is algebraic but, for irrational n, the curve is transcendental.

The curve drawn above is the case n = 4. For even integers n the curve becomes closer to a rectangle as n increases. For odd integer values of n the curve looks like the even case in the positive quadrant but goes to infinity in both the second and fourth quadrants.

The case n = 2/3 is the Astroid, while the case n = 3 is the (so-called) Witch of Agnesi.

The case n = 5/2 has been associated with the Danish architect and poet Piet Hein (inventor of the Soma Cube) under the name of superellipse and have been used for a variety of purposes, including motorway bridges and other architectural applications.

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JOC/EFR/BS January 1997

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