Dumbbell Curve

Dumbbell Curve DUMBBELL CURVE

Balmoral Software

The dumbbell curve is a bisymmetric sextic curve S with parametric equations

x(t) = cos(t)
y(t) = cos²(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) = 3cos³(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

x²(t) + y²(t) = cos²(t) + cos⁴(t)sin²(t)

is 1, so that is the circumradius.

Circumellipse

From Lemma B, we have

x(t)y(t) = cos³(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]cos²(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

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:

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