 Department of Mathematics Education

# Some Explorations with Lemniscates.

### 1. Lemniscate of Bernoulli  ### Explorations:

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  ### Explorations

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.

End