Bowtie
**BOWTIE**

*Balmoral Software*

The bowtie is
a figure-8 curve with polar equation
p(t) = 2 sin(2t) + 1, 0 ≤ t < 2π

Rotating the curve clockwise by π/4 produces the equivalent bisymmetric curve
S with polar equation
r(t) = p(t + π/4) = 2 cos(2t) + 1, 0 ≤ t < 2π

The corresponding implicit function can be found by replacing cos(t) with x/r
and r with x^{2} + y^{2}:
(x^{2} + y^{2})^{3} = (3x^{2} - y^{2})^{2}

The corresponding parametric functions are:
x(t) = 2cos(t) + cos(3t)
y(t) = sin(3t)

Movement along S is always counterclockwise, crossing the origin four times. At
t = 0, the path starts at the maximum abscissa point (3,0) and
moves through Quadrant I along its right lobe to its maximum ordinate point
at t = π/6, and
then on to the origin at t = π/3. It continues along the lower
small lobe, returning to the origin at t = 2π/3. Next, it
travels along the left lobe through Quadrant II to another maximum ordinate
point at
t = 5π/6, then on to its minimum abscissa point (-3,0) at
t = π and back to the origin at t = 4π/3. The
path continues through the upper small lobe, including a third maximum ordinate
point (0,1) at t = 3π/2, the origin and Quadrant IV back to its
starting point at t = 2π:
**t:** | 0 | π/6 | π/3 | π/2 | 2π/3 | 5π/6 | π | 7π/6 | 4π/3 | 3π/2 | 5π/3 | 11π/6 | 2π |

**x(t):** | 3 | | 0 | 0 | 0 | - | -3 | - | 0 | 0 | 0 | | 3 |

**y(t):** | 0 | 1 | 0 | -1 | 0 | 1 | 0 | -1 | 0 | 1 | 0 | -1 | 0 |

Since S has three maximum-ordinate points, it is non-convex by the
multiple local extrema test. The
width x height of the bowtie's bounding rectangle is 6 x 2.
### Metrics

We have
r'(t) = -4 sin(2t),

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
each of which is bisected by
another extreme ordinate point on one of the small lobes. By
(L2), the perimeter of the convex hull is

which is about 30% shorter than that of the bowtie.
The line segments of the convex hull create two isosceles triangles with the
origin, each having an area of

as shown in blue in the left diagram below. By
(A2), the area of the convex hull is

which is about 16% more than the area of the bowtie.
### Circumcircle

The radius maximum is 3, so that is the circumradius.
### Circumellipse

From Lemma B, we have
x(t)y(t) = [2cos(t) + cos(3t)]sin(3t)

This expression is maximized in the first quadrant at the complicated value

The circumellipse dimensions are

For verification, we have

### Incircle (lobe)

Consider the right lobe of the bowtie, where -π/3 ≤ t < π/3. The
maximum radius of a circle centered on the x-axis and inscribed in the lobe is
the maximum ordinate 1 of S, so a candidate for the lobe incircle has radius
R = 1 and center abscissa
The circle must be contained
within the lobe, so we require that c - R and c + R
both be within the abscissa range [0,3] of the lobe, which is true. For
verification, we have

### Inellipse (lobe)

Using z = 3 in Lemma E,
d/dt [x(t) - z]y(t) = 2 cos(2t) - 9 cos(3t) + 4 cos(4t) + 3 cos(6t)

has a zero at t* = 0.711832. The corresponding coordinates are
x* = 0.979171
y* = 0.844749

We then have

For verification, we have

### Summary Table

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

Incircle (lobe) | R = 1 | 6.283185 | 3.141593 | (1.732051,0) |

Inellipse (lobe) | a = 1.347219 b = 0.975432 | 7.343639 | 4.128431 | (-1.652781,0) |

Bowtie | Width: 6 Height: 2 | 20.015781 | 9.424778 | |

Convex hull | | 14.098791 | 10.935821 |

Circumellipse | a = 3.262063 b = 1.268192 | 14.930224 | 12.996524 |

Circumcircle | R = 3 | 18.849556 | 28.274334 |

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

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