Department of Mathematics Education

# Some Explorations with Lemniscates.

#### by

Jim Wilson

### 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