Department of Mathematics Education

Some Explorations with Lemniscates.


Jim Wilson

1. Lemniscate of Bernoulli


A. Try replacing the 25 by some other coefficients to observe the effect.

B. Try

for different values of A, A > 0, Include A> 1, A< 1. Note there is no graph when A < 0. Why?

C. Try

for different values of B. Include B > 1, B <1.

What happens if B = 0? if B < 0? Try them!!

D. The geometric definition of the Lemniscate of Bernoulli is the locus of points with the product of the distances from two fixed points (foci) set at a particular constant. Construct such a locus with GSP.

E. Try in polar coordinates. (Note: It will work with Algebra Xpresser).

F. Try

G. Try

H. Try

I. Replace x by (x-1), y by 3y in G.

J. Replace x by sin(x) and y by sin(y) in G. Replace the coefficient 50 by smaller numbers, such as 2, 3, 4 or -1, -2, -3.

2. Lemniscate of Gerone


A. Try

B. Compare the Lemniscate of Bernoulli and the Lemniscate of Gerone.

C. Write an equation for this graph:

D. Write an equation for this graph:

E. Is there a geometric construction, as a locus, for the Lemniscate of Gerone? If so, implement it on GSP.