Epicycloid: How do I get the characteristic equation given a picture and specific points

Question

I am simulating a process and the resulting line has the shape of an epicyloid. the epicycloid- like shape of the line

For my next steps I need an equation to approach this shape. I got the paramatic equation of an epicycloid:

x = (r+R)cos(t)-acos((1+R/r)*t);

y = (r+R)sin(t)-asin((1+R/r)*t);

And I got the specific points. So i got the size of x and y at given t. My consideration was to take 3 specific points in order to solve 3 equations with 3 unknown variables and define R, r and a in this way.

But I can't find a way to solve the equations.

Do one of you got an idea how to get the unknown Parameters ?

Answer 1

You are talking about prolate epicycloid. If you take function $f(t) = x^2 + y^2$:

$$ f(t) = a^2 +(R+r)^2 - 2a(R+r)\cos\left(\frac Rrt\right) = A - B\cos \omega t $$

You can find the period of the function, maximal and minimal value of the function this will give you all the parameters.

In other words, you find the angle $\theta$ between to successive points closest to origin, you find the minimal and maximal distances to origin $D$ and $d$:

$$ R/r = 2\pi/\theta,\qquad D = a+R+r,\qquad d = R+r-a,\\ a = \frac{D+d}2,\qquad R= \frac{D-d}{2+\theta/\pi},\qquad r=\frac{D-d}{4\pi/\theta+2} $$

Answer 2

@Vasily Mitch has produced a theoretical answer.

Here is a practical approach based on a somewhat trial and error basis that can be valuable in order to have intuition on these matters and be sometimes able (with a good understanding) to situate which parameter(s) one has to adjust.

Here is the result :

Let us consider this curve as the absolute trajectory of a "Moon" revolving around an "Earth", itself revolving around a "Sun" placed at the origin.

What did I do ?

I first observed that you had "almost" n=3 loops. Thus you must have "almost" a $n+1:1=4$ ratio for the angular speed of revolving Moon with respect to the angular speed of Earth, taken as unit (in fact, I have taken $3.8$).

Then I used the fact that the max and min distances of the "Moon" with respect to the Sun are resp. $200$ and $40$. Thus the radius of the Earth's trajectory (materialized by a dotted line) must be half-distance $(200+40)/2=120$. Whence, with complex notations (that are more compact) the variable point :

$$z=\color{red}{e^{0.5i}}(120 e^{it}+80e^{i3.8t})\tag{1}$$

Why this factor in red ? In order to produce a last global rotation with $0.5$ radians.

For plotting, take the real and imaginary parts of (1) :

$$x=120 \cos(t+0.5)+80\cos(3.8t+0.5), \ \ \ y= 120 \sin(t+0.5)+80\sin(3.8t+0.5)$$