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) = r^{2}(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 |

| |

*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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