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This bisymmetric curve is traced out in a counterclockwise direction around the origin, starting from the right edge at (1,0). Its maximum height occurs at the points which determine that the dipole is non-convex by the multiple local extrema test. The values of t at these points are and π - . The maximum width of the dipole is between the point pair (0,±1), so the width x height of its bounding rectangle is
so by (L2), the perimeter of S is
and by (A2), the area of S is
which is about 24% shorter than that of the dipole.
The line segments of the convex hull create two isosceles triangles with the origin, each having an area of
By (A2), the area of the convex hull is
which is about 9% more than the area of the dipole.
x(t)y(t) = r2(t)cos(t)sin(t) = |cos(t)|sin(2t)/2This expression is maximized in the first quadrant at The circumellipse dimensions are
For verification, we have
d/dt [x(t) - z]y(t) = d/dt |cos(t)|sin(2t)/2We established in the preceding Circumellipse section that this expression has a zero in the first quadrant at The corresponding coordinates are
We then have
For verification, we have
The value of c is negated in the diagram below to display the inellipse separately in the left lobe.
Figure Parameters Perimeter Area Centroid Incircle (lobe) R = 1/2 3.141593 0.785398 (0.5,0) Inellipse (lobe) 3.446495 0.930842 (0.491859,0) Dipole 7.16557 2 Convex hull 5.460413 2.177324 Circumellipse 5.636992 2.418399 Circumcircle R = 1 6.283185 3.141593
Thong? |
The dipole (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 Lemniscate of Gerono:
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