Epicycloid
**EPICYCLOID**

*Balmoral Software*

A three-lobed
epicycloid is a
closed curve S with parametric equations
x(t) = 4cos(t) - cos(4t)
y(t) = 4sin(t) - sin(4t), 0 ≤ t < 2π

The curve is traced out in a counterclockwise direction, starting from its right
cusp at (3,0). Its abscissa extrema are (-5,0) at t = π and

at t = ±π/5, so the epicycloid is non-convex by the
multiple local extrema test. Its
ordinate extrema are

at t = ±2π/5, so the width x height of its bounding rectangle is

### Symmetry

Since
x(t + 2π/3) = cos(2π/3)x(t) - sin(2π/3)y(t)

and
y(t + 2π/3) = sin(2π/3)x(t) + cos(2π/3)y(t),

the conditions (R1) of Lemma R are
satisfied and the epicycloid is rotationally symmetric with period 2π/3.
### Metrics

We have
x'(t) = 4[sin(4t) - sin(t)]
y'(t) = 4[cos(t) - cos(4t)],

so by (L1), the perimeter of S is

and by (A1), the area of S is

### Convex Hull

The convex hull is created by connecting the maximum abscissa points with a
vertical line segment of length
and then repeating the pattern by rotating around the origin at 2π/3
intervals. By
(L2), the perimeter of the convex hull is

which is about 8% shorter than that of the epicycloid.
To compute the area of the convex hull, we can see by
(A1), that the area A_{1} of the
region in solid green in the left diagram below is the integral of
y dx from x = 3 at the cusp to
at the maximum abscissa point:

Therefore, the area of the convex hull is

which is about 4% more than the area of the epicycloid.
### Boundary Circles & Circumellipse

Since the epicycloid is rotationally symmetric with period 2π/3, its
circumcircle and circumellipse are the same. The squared-distance function of S
x^{2}(t) + y^{2}(t) = 17 - 8cos(3t)

ranges from 9 to 25, so its inradius is 3 and its circumradius is 5.
### Inellipse

The lateral asymmetry of S permits an offset inellipse centered on the x-axis
and tangent at all three cusps. The search program produces the optimal result
a = 2.453277

b = 4.712299

c = 0.546723

Perimeter = 23.074230

Area = 36.318614

### Lobe Circumellipse

It's interesting that a lobe of the epicycloid is almost perfectly elliptical in
shape, as indicated by the close-fitting brown ellipse in the left diagram
below. The height of this x-symmetric ellipse is the maximum ordinate of S over
the lobe parameter range 2π/3 ≤ t ≤ 4π/3:

The corresponding abscissa is the center of the ellipse:

For tangency at the left edge (-5,0), we have

To verify that this ellipse encloses the lobe of S, we have

The ellipse appears to pass through the points (0,±1), but this an
illusion due to the limited resolution of the image, since

rather than 1. Although the ellipse is close-fitting, it is not the
minimum-area circumellipse of the lobe, which can be found by numerical search:
a = 2.269858

b = 3.112156

c = -2.730142 = a - 5

and is shown in green in the left diagram below.
### Summary Table

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

Lobe ellipse | a = b = | 17.335157 | 23.755802 | (-2.427051,0) |

Incircle | R = 3 | 18.849556 | 28.274334 | |

Inellipse | a = 2.453277 b = 4.712299 | 23.074230 | 36.318614 | (0.546723,0) |

Epicycloid | | 32 | 62.831853 | |

Convex hull | | 29.389263 | 65.552643 |

Circumcircle | R = 5 | 31.415927 | 78.539816 |

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