Lemniscate of Gerono
**LEMNISCATE OF GERONO**

*Balmoral Software*

The Lemniscate of
Gerono is a bisymmetric figure-eight curve S with parametric equations
x(t) = sin(t)
y(t) = sin(2t)/2, 0 ≤ t < 2π

The path of S starts at the origin, passes clockwise through Quadrants I and
IV, then counterclockwise through Quadrants II and III and back to the origin.
Its abscissa extrema are on the x-axis at ±1 when
t = ±π/2, and its extreme ordinate points are
at
t = ±π/4, ±3π/4, so the width x height of its
bounding rectangle is 2 x 1.
### Metrics

The perimeter of S is 6.097223 (OEIS
A118178) and its area is 4/3.
### Convex Hull

The convex hull is created by connecting the extreme ordinate points with two
horizontal line segments of length
. We have
x'(t) = cos(t)
y'(t) = cos(2t),

so by (L1), the perimeter of the convex
hull is

which is about 12% shorter than that of the lemniscate.
The line segments of the convex hull create a rectangle of area
, as shown in blue in the left
diagram below. By (A1), the area of the
convex hull is

which is about 41% more than the area of the lemniscate.
### Circumcircle

The maximum squared distance
x^{2}(t) + y^{2}(t) = sin^{2}(t) + sin^{2}(2t)/4

is 1, so that is the circumradius.
### Circumellipse

From Lemma B, we have
x(t)y(t) = sin(t)sin(2t)/2

This expression is maximized in the first quadrant at
so the circumellipse dimensions
are

For verification, we have

### Incircle (lobe)

Consider the right lobe of the lemniscate, where 0 ≤ t < π. The maximum
ordinate 1/2 of the lobe does not define its inradius since the corresponding
abscissa is too close to the
right edge (1,0), so its incircle is constrained by the right edge of S. Using
z = 1 in Lemma C, we have

and R = |c - z| = 11/27. For verification, we have

### Inellipse (lobe)

Using z = 1 in Lemma E,
d/dt [x(t) - z]y(t) = [sin(t) - 1]sin(2t)/2

has a zero at

The corresponding coordinates are

We then have

For verification, we have

### Summary Table

**Figure** | **Parameters** | Perimeter | Area | Centroid |

Incircle (lobe) | R = 11/27 | 2.559816 | 0.521444 | (0.592593,0) |

Inellipse (lobe) | | 2.609178 | 0.535201 | (0.622839,0) |

Lemniscate of Gerono | Width: 2 Height: 1 | 6.097223 | 1.333333 | |

Convex hull | | 5.392483 | 1.885618 |

Circumellipse | | 5.825171 | 2.418401 |

Circumcircle | R = 1 | 6.283185 | 3.141593 |

| |

*Binoculars* | |

The Lemniscate of Gerono (red) is a member of a group of figure-8 curves
described on these pages, including (inside to outside) the
dumbbell curve, the
bowtie, the
Lemniscate of Bernoulli and the
dipole:

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