Dumbbell Curve
**DUMBBELL CURVE**

*Balmoral Software*

The dumbbell curve
is a bisymmetric sextic curve S with parametric equations
x(t) = cos(t)
y(t) = cos^{2}(t)sin(t), 0 ≤ t < 2π

The curve follows a counterclockwise path around the origin, starting from its
right edge at (1,0). Its extreme abscissa points are (±1,0) and its
maximum ordinate point in the first quadrant is
so the width x height of its
bounding rectangle is
### Metrics

The perimeter of S is 5.541003 and its area is π/4.
### Convex Hull

The convex hull is created by connecting the extreme ordinate points with two
horizontal line segments of length
We have
x'(t) = -sin(t)
y'(t) = 3cos^{3}(t) - 2cos(t),

so by (L1), the perimeter of the convex
hull is

which is about 7% shorter than the perimeter of the dumbbell curve.
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 almost twice the area of the dumbbell curve.
### Circumcircle

The maximum squared distance
x^{2}(t) + y^{2}(t) = cos^{2}(t) + cos^{4}(t)sin^{2}(t)

is 1, so that is the circumradius.
### Circumellipse

From Lemma B, we have
x(t)y(t) = cos^{3}(t)sin(t)

This expression is maximized in the first quadrant at t* = π/6, so the
circumellipse dimensions are

For verification, we have

### Incircle (lobe)

Consider the right lobe of the dumbbell curve, where -π/2 ≤ t < π/2.
The maximum ordinate of the
lobe does not define its inradius since the corresponding abscissa
is too close to the right edge
(1,0), so its incircle is constrained by the right edge of S. Using z = 1 in
Lemma C,

is an extremely complicated expression approximated by 0.704642. The
corresponding radius R = |c - z| = 0.295358. For verification, we
have

### Inellipse (lobe)

Using z = 1 in Lemma E,
d/dt [x(t) - z]y(t) = d/dt [cos(t) - 1]cos^{2}(t)sin(t)

has a zero at

The corresponding coordinates are

We then have

For verification, we have

### Summary Table

**Figure** | **Parameters** | Perimeter | Area | Centroid |

Incircle (lobe) | R = 0.295358 | 1.855791 | 0.274062 | (0.704642,0) |

Inellipse (lobe) | | 1.883674 | 0.278699 | (0.728714,0) |

Dumbbell curve | | 5.541003 | 0.785398 | |

Convex hull | | 5.135534 | 1.885618 |

Circumellipse | | 5.731721 | 2.040524 |

Circumcircle | R = 1 | 6.283185 | 3.141593 |

| |

*Peripheral Vision* | |

The dumbbell curve (red) is a member of a group of figure-8 curves described on
these pages, including (inside to outside) the
bowtie, the
Lemniscate of Bernoulli, the
Lemniscate of Gerono and the
dipole:

Top Page

Home

Copyright © 2021 Balmoral Software (http://www.balmoralsoftware.com). All
rights reserved.