Mouth Curve
**MOUTH CURVE**

*Balmoral Software*

The "Mouth
curve" is a bisymmetric sextic curve S with parametric equations
x(t) = cos(t)
y(t) = sin^{3}(t), 0 ≤ t < 2π

The curve is traced out in a counterclockwise direction, starting from its
right-hand cusp at (1,0). Its abscissa extrema are (±1,0) and its
ordinate extrema (0,±1), so its bounding rectangle is a 2 x 2 square.
### Metrics

We have
x'(t) = -sin(t)
y'(t) = 3sin^{2}(t)cos(t),

so by (L1), the perimeter of S is

By (A1), the area enclosed by S is

### Convex Hull

We have
x''(t) = -cos(t)
y''(t) = 6sin(t) - 9sin^{3}(t),

so by (X1), the
curvature of S changes sign when t = 0 since that value is a
solution to
-sin(t)[6sin(t) - 9sin^{3}(t)] = -cos(t)[3sin^{2}(t)cos(t)]

Equation (X2) is
evaluated with respect to the point (x(0),y(0)) = (1,0) as

which has a solution in the first quadrant at t = π/3. The
convex hull is created by connecting the points
and (1,0) with a straight line
segment of length and
symmetrically in the other three quadrants. By
(L1), the perimeter of the convex hull is:

which is about 0.4% shorter than the perimeter of S.
In the left figure below, the area of the Quadrant I portion of the triangle
outlined in blue is

and the area of the remaining convex hull area in Quadrant I is

so the total area of the convex hull is

which is about 2% larger than the area of S.
### Boundary Circles & Circumellipse

The squared distance function x^{2}(t) + y^{2}(t)
ranges from to 1, so the
inradius is

and the circumradius is 1. Since the maximum distance of S from the origin
along its positive and negative axes is the same, its circumellipse is the same
as its circumcircle.
### Inellipse

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

is maximized in the first quadrant when t^{*} = π/3, so the inellipse
dimensions are

For verification, we have

### Summary Table

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

Incircle | R = | 4.927795 | 1.932393 | |

Inellipse | | 5.128805 | 2.040524 |

Mouth curve | Width: 2 Height: 2 | 5.82896 | 2.356195 |

Convex hull | | 5.805229 | 2.409196 |

Circumcircle | R = 1 | 6.283185 | 3.141593 |

| |

*The Aria* | |

The mouth curve (red) is a member of a group of similarly-shaped figures
described on these pages, including (inside to outside) the
Vesica Piscis and the
cycloid:

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