Following our previous experiments with cycloids around circles, we now move into a new space - cycloids around ellipses.
It would seem a logical progression, but has presented some difficulties.
Foremost is the formula of the cycloid - there doesn't appear to be one.
However, as shown above, the blue dot does trace an epicycloid around the ellipse. Now all we must do is determine the formula to produce it.
Another problem is the number of revolutions around the ellipse.
For this we need 2 values:
Radius r = 30
Circumference (c) = 2𝛑r
≅ 188.5
Calculating the perimeter of the ellipse is somewhat difficult. There are several approximate solutions, as well as 'exact' solutions.
Further readings:
We will use one from the first link above:
C = 𝛑[3(a + b) - √(3a+b)(a+3b)]
where a = semi-major axis and b = semi-minor axis.
Our example above holds the following values:
A = 240, a = 120
B = 180, b = 90
C = 𝛑 [3(120 + 90) - √(450*490)]
= 𝛑 [630 - √220500]
= 𝛑 [630 - 469.58]
= 𝛑 * 160.43
≅ 504.0
Knowing both we can now determine how many cusps our epicycloid will have.
504 / 188.5 ≅ 2.67
However a crucial value is the speed of rotation vs speed of revolution.
Next we show an elliptical hypocycloid ...