cykloida

$\varphi=%v$

r  (radius of the circle)

$\textstyle x=(R+r)\cos{\varphi}-c\cos{\frac{(R+r)\varphi}{r}}$, $\textstyle y=(R+r)\sin{\varphi}-c\sin{\frac{(R+r)\varphi}{r}}$, $\varphi\in R$

K

$φ$

$\text{Angle }φ$

$\text{Circle }K$

$\text{Circle }k$

$\text{Epicycloid [shadow]}$

$\text{Point }B$

$\text{Epicycloid}$

R = 2 (slider from 0 to (10 * r), step 0.1)
φ = 6.283185307179586 (slider from 0 to ((20 * r / d) * π), step 0.1)
r = 1 (slider from 0.1 to 5, step 0.1)
c = 1 (slider from 0 to (5 * r), step 0.1)
S: (((R + r) * cos(φ)), ((R + r) * sin(φ)))
k = Circle(S, r)
B: (((R + r) * cos(φ)) - (c * cos(((R + r) * φ / r))), ((R + r) * sin(φ)) - (c * sin(((R + r) * φ / r))))
f = Segment(B, S)
h = Ray(S, B)
i = Circle((0, 0), R)
d = GCD((10 * R), (10 * r))
j = Segment((0, 0), S)
A = Intersect(k, h)
g = Segment(A, S)
α = Angle((R, 0), (0, 0), S)
a = CurveCartesian(((R + r) * cos(tt)) - (c * cos(((R + r) * tt / r))), ((R + r) * sin(tt)) - (c * sin(((R + r) * tt / r))), tt, 0, φ)
b = CurveCartesian(((R + r) * cos(tt)) - (c * cos(((R + r) * tt / r))), ((R + r) * sin(tt)) - (c * sin(((R + r) * tt / r))), tt, 0, ((20 * r / d) * π))
text2: "Epicycloid " + (LaTeX[If[r ≟ c, "\text{ Ordinary }(c=r)", If[r > c, "\text{ Curtate }(c<r)", "\text{ Prolate }(c>r)"]]]) + ""
e = true
l = true
m = true
n = true
o = true
p = true
Rr: R + r
text1_1: "$\textstyle x=" + (LaTeX[Rr]) + "\cos{\varphi}-" + (LaTeX[c]) + "\cos{\frac{" + (LaTeX[Rr]) + "\varphi}{" + (LaTeX[r]) + "}}$, $\textstyle y=" + (LaTeX[Rr]) + "\sin{\varphi}-" + (LaTeX[c]) + "\sin{\frac{" + (LaTeX[Rr]) + "\varphi}{" + (LaTeX[r]) + "}}$, $\varphi=" + (LaTeX[φ]) + "$"