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The hippopede, also known as the Lemniscate of Booth, is a peanut-shaped bisymmetric curve S with polar equation
r2(t) = 4 - 3sin2(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


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


From Lemma B, we have
x(t)y(t) = [4 - 3sin2(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

IncircleR = 16.2831853.141593
Inellipsea = 2
b = 1
Convex hull10.6221068.076415
CircumcircleR = 212.56637112.566371

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