Hippopede
**HIPPOPEDE**

*Balmoral Software*

The hippopede, also known as
the Lemniscate of Booth, is a peanut-shaped bisymmetric curve S with polar
equation
r^{2}(t) = 4 - 3sin^{2}(t), 0 ≤ t < 2π

The curve is traced out in a counterclockwise direction, starting from its right
edge at (2,0). Its abscissa extrema are (±2,0) and its ordinate extrema
so the hippopede is non-convex
by the multiple local extrema
test. In the first quadrant, the ordinate maximum is achieved when
The width x height of its
bounding rectangle is
### Metrics

We have

so by (L2), the perimeter of S is

and by (A2), the area of S is

### Convex Hull

The convex hull is created by connecting the extreme ordinate points with two
horizontal line segments of length
so by
(L2), the perimeter of the convex hull is

which is less than 1% shorter than that of the hippopede.
The line segments of the convex hull create two isosceles triangles with the
origin, each having an area of

as shown in blue in the left diagram below. By
(A2), the area of the convex hull is

which is about 3% more than the area of the hippopede.
### Boundary Circles & Inellipse

The radius extrema are on the coordinate axes at (0,±1) and
(±2,0), so
the inradius is 1 and the circumradius is 2. A candidate for the inellipse is
one enclosed by the annulus between these two circles, with a = 2
and b = 1. For verification, we have

### Circumellipse

From Lemma B, we have
x(t)y(t) = [4 - 3sin^{2}(t)]cos(t)sin(t)

This expression is maximized in the first quadrant at the complicated value

which is close to The
circumellipse dimensions are

For verification, we have

### Summary Table

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

Incircle | R = 1 | 6.283185 | 3.141593 | |

Inellipse | a = 2 b = 1 | 9.688448 | 6.283185 |

Hippopede | | 10.6941 | 7.853982 |

Convex hull | | 10.622106 | 8.076415 |

Circumellipse | | 10.955915 | 8.926023 |

Circumcircle | R = 2 | 12.566371 | 12.566371 |

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