Cayley Sextic
**CAYLEY SEXTIC**

*Balmoral Software*

The Cayley sextic
having its cusp on the right is a heart-shaped closed curve S with polar
equation
r(t) = -2cos^{3}(t/3), -pi ≤ t < pi

For a simple curve describing the outer envelope of the Cayley sextic, T is
restricted to the interval [-π,π]. The scaling factor -2 is used for
comparison with the cardioid. Since r(t)
is an even function of t, the Cayley sextic is symmetric with respect to the
x-axis.
The curve is traced out in a counterclockwise direction, starting from the cusp
(1/4,0) at t = -π. Its minimum abscissa occurs at the point
(-2,0) when t = 0. Its maximum abscissas occur at the points
(1/2,±1/2) when t = ∓3π/4, which determine that
the Cayley sextic is non-convex by the
multiple local extrema test. The
extreme ordinates of the Cayley sextic occur at the points
when
t = ∓3π/8, so the width x height of its bounding
rectangle is

### Metrics

We have
r'(t) = 2sin(t/3)cos^{2}(t/3),

so by (L2), the perimeter of this portion
of the Cayley sextic is

By (A2), the corresponding area is

By (C2), the centroid abscissa of
this portion of the Cayley sextic is

The bounding rectangle for inconics is delimited by the cusp at (1/4,0).
### Convex Hull

The convex hull is created by connecting the maximum abscissa points with a
vertical line segment of length 1 at x = 1/2. By
(L2), the perimeter of the convex hull is

which is about 2% shorter than that of the Cayley sextic portion.
The line segment of the convex hull creates an isosceles triangle with the
origin, having area (1/2)^{2} = 1/4, as shown in blue in
the left diagram below. The centroid abscissa of this triangle is the average
of its vertex abscissas, or 1/3. By (A2),
the remainder of the convex hull has area

By (C2), the associated centroid
abscissa is

The convex hull component metrics can be summarized as follows:
**Region** | Area | Centroid abscissa | Product |

Triangle | 1/4 | 1/3 | 1/12 |

Remainder | A_{1} = (15π + 44)/16 | -(315π + 1016)/(480A_{1}) | -(315π + 1016)/480 |

Total | 15π/16 + 3 | | -(315π + 976)/480 |

The area of the convex hull is about 1% larger than that of the Cayley sextic.
The centroid abscissa of the convex hull is the
weighted
average

### Incircle

A candidate incircle is located on the x-axis between the minimum abscissa -2
and the cusp (1/4,0), and so has radius R = 9/8 and center abscissa
c = -7/8. For verification, we have

### Inellipse

Using z = 1/4 in Lemma E,
d/dt [x(t) - z]y(t) = d/dt 4sin(t)cos(t)cos^{6}(t/3) + sin(t)cos^{3}(t/3)/2

has a zero at t* = -0.707452. The corresponding coordinates are
x* = -1.39729
y* = 1.19484

We then have

For verification,

### Circumellipse

Using z = -2 in Lemma E,
d/dt [x(t) - z]y(t) = d/dt 4sin(t)cos^{6}(t/3)cos(t) - 4sin(t)cos^{3}(t/3)

has a zero at t* = -1.598518. The corresponding coordinates are
x* = 0.035429
y* = 1.2777

We then have

For verification,

### Circumcircle

The radius of a circle centered on the x-axis and circumscribing S is at least
its maximum ordinate, so a candidate for the circumcircle has radius
and center abscissa
For verification,

### Summary Table

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

Incircle | R = 9/8 | 7.068584 | 3.976078 | (-0.875,0) |

Inellipse | a = 1.098193 b = 1.379682 | 7.809609 | 4.760006 | (-0.848193,0) |

Cayley sextic | | 8.881262 | 5.875548 | (-0.702073,0) |

Convex hull | | 8.712389 | 5.945243 | (-0.688787,0) |

Circumellipse | a = 1.356954 b = 1.475361 | 8.901868 | 6.289459 | (-0.643046,0) |

Circumcircle | R = | 9.155272 | 6.670104 | (-0.603553,0) |

The Cayley sextic (red) is a member of a group of similarly-shaped figures
described on these pages, including (inside to outside) the rotated
lima bean curve, the
cardioid and the
cochleoid:

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