Nephroid
**NEPHROID**

*Balmoral Software*

The nephroid is a two-cusped
epicycloid S with parametric equations
x(t) = 3cos(t) - cos(3t)
y(t) = 3sin(t) - sin(3t), 0 ≤ t < 2π

This bisymmetric curve is traced out in a counterclockwise direction around the
origin, starting from the cusp at (2,0). Its maximum width occurs at the point
pairs which determine that the
nephroid is non-convex by the
multiple local extrema test. The
values of t at these point pairs are ±π/4 (right pair) and
±3π/4 (left pair). The maximum height of the nephroid is between the
point pair (0,±4), so the width x height of its bounding rectangle is
### Metrics

The perimeter of the nephroid is 24 and its area A = 12π.
### Convex Hull

The convex hull is created by connecting the extreme abscissa points with
vertical line segments of length
as shown in blue in the left
diagram below. We have
x'(t) = -3sin(t) + 3sin(3t)
y'(t) = 3cos(t) - 3cos(3t)

so by (L1), the perimeter of the convex
hull is

which is about 6% shorter than that of the nephroid.
To compute the area of the convex hull, we can see by
(A1) that the area A_{1} of the
region in green in the left diagram below is the integral of y dx from
x = 2 at the cusp to
at the maximum abscissa point:

Therefore, the area of the convex hull is
A + 4A_{1} = 12π + 4(5 - 3π/2) = 20 + 6π = 38.849556,

which is about 3% more than the area of the nephroid.
### Boundary Circles & Inellipse

The squared-distance function of S
x^{2}(t) + y^{2}(t) = 10 - 6cos(2t)

has its extrema on the coordinate axes at (±2,0) and (0,±4), so
the inradius is 2 and the circumradius is 4. A candidate for the inellipse is
one enclosed by the annulus between these two circles, with a = 2
and b = 4. For verification, we have

### Circumellipse

From Lemma B,
x(t)y(t) = [9sin(2t) - 6sin(4t) + sin(6t)]/2

is maximized in the first quadrant when

The circumellipse dimensions are

For verification, we have

### Summary Table

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

Incircle | R = 2 | 12.566371 | 12.566371 | |

Inellipse | a = 2 b = 4 | 19.376896 | 25.132741 |

Nephroid | | 24 | 37.699112 |

Convex hull | | 22.627417 | 38.849556 |

Circumellipse | | 23.181126 | 41.848410 |

Circumcircle | R = 4 | 25.132741 | 50.265482 |

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