Cycloid
**CYCLOID**

*Balmoral Software*

Consider the closed convex shape S created from one
cycloid arch, translated to the
left by π and joined to its reflection across the x-axis. So that S is
traversed in a counterclockwise direction around the origin starting from its
right-hand corner (π,0) at t = 0, its parametric equations are

Since y(π - t) = y(t) and the path of the lower arch is a
point
reflection through the origin of the upper arch, S is bisymmetric. Its
abscissa extrema are (±π,0) and its ordinate extrema (0,±2), so
the width x height of its bounding rectangle is 2π x 4. Its perimeter is
double that of one cycloid arch: 2(8) = 16, and similarly, its area
is 2(3π) = 6π.
### Boundary Circles & Circumellipse

The squared distance function x^{2}(t) + y^{2}(t)
of S ranges from a minimum of 4 at (0,±2) to a maximum of
π^{2} at (±π,0), so the inradius is 2 and the circumradius
is π. Since these extrema occur on the coordinate axes, a candidate for the
circumellipse is one enclosed by the annulus between the two boundary circles,
with a = π and b = 2. For verification, we have

### Inellipse

In Lemma B, we have for the upper arch
x(t)y(t) = [-2t + π + sin(2t)][1 - cos(2t)]

This expression is maximized at
t* = 0.984995,

so the inellipse dimensions are

For verification, we have

### Summary Table

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

Incircle | R = 2 | 12.566371 | 12.566371 | |

Inellipse | a = 2.959917 b = 1.963884 | 15.627233 | 18.261870 |

Cycloid arches | Width: 2π Height: 4 | 16 | 18.849556 |

Circumellipse | a = π b = 2 | 16.352486 | 19.739211 |

Circumcircle | R = π | 19.739209 | 31.006277 |

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

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