Lemniscate of Bernoulli
**LEMNISCATE OF BERNOULLI**

*Balmoral Software*

The Lemniscate of
Bernoulli, or two-leaved rose, is a bisymmetric figure-8 curve S with
parametric equations

for 0 ≤ t < 2π. As t increases from 0, S starts from its right-hand edge
at (1,0), moves through Quadrant I to the origin at t = π/2,
then clockwise through Quadrant III, its left edge (-1,0) at
t = π, and Quadrant II before returning to the origin at
t = 3π/2 and passing through Quadrant IV back to its starting
point. The ordinate extrema of S are at
when
so the width x height of its
bounding rectangle is
### Metrics

The perimeter of S is

and its area is 1.
### Convex Hull

The convex hull is created by connecting the ordinate extrema points with two
horizontal line segments of length
We have

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

which is about 9% 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 30% more than the area of the lemniscate.
### Circumcircle

The squared-distance function of S

is maximized at 1, so the circumradius is 1.
### Circumellipse

From Lemma B, we have

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 -π/2 ≤ t < π/2. The
maximum radius of a circle centered on the x-axis and inscribed in the lobe is
the maximum ordinate of S, so a
candidate for the lobe incircle has radius
and center abscissa
The candidate circle must be
contained within the lobe, so we require that c - R and
c + R both be within the abscissa range [0,1] of the lobe, which is
true. For verification, we have

### Inellipse (lobe)

Using z = 1 in Lemma E,

has a zero at t* = 0.916671. The corresponding coordinates are
x* = 0.373344
y* = 0.296279

We then have

For verification,

### Summary Table

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

Incircle (lobe) | R = | 2.221442 | 0.392699 | (0.612372,0)q |

Inellipse (lobe) | a = 0.417771 b = 0.342114 | 2.393169 | 0.449013 | (-0.582229,0) |

Lemniscate of Bernoulli | | 5.244115 | 1 | |

Convex hull | | 4.785822 | 1.299038 |

Circumellipse | | 5.093049 | 1.570797 |

Circumcircle | R = 1 | 6.283185 | 3.141593 |

| |

*Bug-eyed Space Jockey* | |

The Lemniscate of Bernoulli (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 Gerono and the
dipole:

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