Tschirnhausen Cubic
**TSCHIRNHAUSEN CUBIC**

*Balmoral Software*

The lobe of the
Tschirnhausen cubic
is a convex x-symmetric droplet-shaped closed curve S with polar equation
r(t) = sec^{3}(t/3), -π ≤ t < π

The curve follows a counterclockwise path as t increases, starting from the cusp
(-8,0) at t = -π and proceeding through Quadrants III and IV to
the right edge (1,0) at t = 0, then following a symmetric path
through Quadrants I and II back to the cusp at t = π. The
maximum ordinate of the Tschirnhausen cubic occurs at (-2,2) when
t = 3π/4, so the width x height of its bounding rectangle is 9 x
4.
### Metrics

We have
r'(t) = tan(t/3)sec^{3}(t/3),

so by (L2), the perimeter of the
Tschirnhausen cubic is

and by (A2), its area A is

By (C2), its centroid abscissa is

### Incircle

The maximum radius of a circle centered on the x-axis and inscribed in S is its
maximum ordinate, so a candidate for the incircle has radius R = 2
and center abscissa c = -2. The candidate circle must be contained
within S, so we require that c - R and c + R both be
within the abscissa range [-8,1] of S, which is true. For verification, we have

### Inellipse

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

has a zero at t* = (3/2)Arccos(-2/7). The corresponding coordinates are

We then have

For verification,

### Circumellipse

Using z = -8 in Lemma E,
d/dt [x(t) - z]y(t) = d/dt [sec^{3}(t/3)cos(t) + 8]sec^{3}(t/3)sin(t)

has a zero at

The corresponding coordinates are

We then have

For verification,

### Circumcircle

Since the abscissa extrema are on the x-axis and the bounding rectangle is wider
than it is tall, a candidate for the circumcircle has radius 9/2 and center
abscissa -7/2. For verification, we have

### Summary Table

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

Incircle | R = 2 | 12.566371 | 12.566371 | (-2,0) |

Inellipse | | 17.588970 | 21.025157 | (-2.6,0) |

Tschirnhausen cubic | Width: 9 Height: 4 | 20.784610 | 24.941532 | (-2.857143,0) |

Circumellipse | | 22.627006 | 32.370342 | (-3.2,0) |

Circumcircle | R = 9/2 | 28.274334 | 63.617251 | (-3.5,0) |

The Tschirnhausen cubic (red) is a member of a group of similarly-shaped figures
described on these pages, including (inside to outside) the
teardrop curve, the
right strophoid, the
Trisectrix of Maclaurin and the
piriform curve:

Top Page

Home

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