Dipole

Dipole DIPOLE

Balmoral Software

The dipole is a figure-8 curve S with polar equation

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

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

as shown in blue in the left diagram below. By (L2), the perimeter of the convex hull 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.

Circumcircle

The radius maximum is 1, so that is the circumradius.

Circumellipse

From Lemma B, we have

x(t)y(t) = r²(t)cos(t)sin(t) = |cos(t)|sin(2t)/2

This expression is maximized in the first quadrant at

The circumellipse dimensions are

For verification, we have

Incircle (lobe)

Consider the right lobe of the dipole, where -π/2 ≤ t < π/2. The lobe width 1 is smaller than its height

so a candidate for a circle inscribed in the lobe is one with a radius R and center abscissa c both equal to 1/2. For verification, we have

Inellipse (lobe)

Using z = 0 in Lemma E,

d/dt [x(t) - z]y(t) = d/dt |cos(t)|sin(2t)/2

We 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.

Summary Table

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

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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