Right Strophoid
**RIGHT STROPHOID**

*Balmoral Software*

The lobe of the
right strophoid
is a convex x-symmetric droplet-shaped closed curve S with polar equation
r(t) = 2cos(t) - sec(t), -π/4 ≤ t < π/4

The curve follows a counterclockwise path as t increases, starting from the cusp
at the origin when t = -π/4, proceeding through Quadrant IV to
the right edge (1,0) at t = 0, then following a symmetric path
through Quadrant I back to the cusp at t = π/4. The maximum
ordinate point of S is

so the width x height of its bounding rectangle is

### Metrics

The area of S is
A = (4 - π)/2

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

so by (L2), the perimeter of S 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

and center abscissa The
candidate circle must be contained within S, so we require that
c - R and c + R both be within the abscissa range
[0,1] 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 -2sin^{2}(t)cos(2t)tan(t)

has a zero at

The corresponding coordinates are

We then have

For verification,

### Circumellipse

Using z = 0 in Lemma E,
d/dt [x(t) - z]y(t) = d/dt cos^{2}(2t)tan(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 its radius and center
abscissa both equal to 1/2. For verification, we have

### Summary Table

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

Incircle | R = | 1.886734 | 0.283277 | (0.618034,0) |

Inellipse | | 2.252459 | 0.382312 | (0.577350,0) |

Right strophoid | | 2.4896 | 0.429204 | (0.553264,0) |

Circumellipse | | 2.668302 | 0.517274 | (0.520518,0) |

Circumcircle | R = 1/2 | 3.141593 | 0.785398 | (0.5,0) |

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

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