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
and its area is 1.
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.
is maximized at 1, so the circumradius is 1.
This expression is maximized in the first quadrant at
so the circumellipse dimensions are
For verification, we have
has a zero at t* = 0.916671. The corresponding coordinates are
x* = 0.373344We then have
y* = 0.296279
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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