cykloida

r  (radius of the circle)

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

K

t

$\text{Angle }t$

$\text{Circle }K$

$\text{Circle }k$

$\text{Epicycloid [shadow]}$

$\text{Point }B$

$\text{Epicycloid}$

R = 2 (slider from 0 to (10 * r), step 0.1)
t = 11.6 (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((r * t / R))), ((R + r) * sin((r * t / R))))
k = Circle(S, r)
B: (((R + r) * cos((r * t / R))) - (c * cos(((R + r) * t / R))), ((R + r) * sin((r * t / R))) - (c * sin(((R + r) * t / R))))
f = Segment(B, S)
h = Ray(S, B)
i = Circle((0, 0), R)
a = CurveCartesian(((R + r) * cos((r * tt / R))) - (c * cos(((R + r) * tt / R))), ((R + r) * sin((r * tt / R))) - (c * sin(((R + r) * tt / R))), tt, 0, t)
d = GCD((10 * R), (10 * r))
b = CurveCartesian(((R + r) * cos((r * tt / R))) - (c * cos(((R + r) * tt / R))), ((R + r) * sin((r * tt / R))) - (c * sin(((R + r) * tt / R))), tt, 0, ((20 * R / d) * π))
j = Segment((0, 0), S)
α = Angle((0, 0), S, B)
A = Intersect(k, h)
g = Segment(A, S)
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: "$\textstyle x=" + (LaTeX[Rr]) + "\cos{\frac{" + (LaTeX[r]) + "t}{" + (LaTeX[R]) + "}}-" + (LaTeX[c]) + "\cos{\frac{" + (LaTeX[Rr]) + "t}{" + (LaTeX[R]) + "}}$, $\textstyle y=" + (LaTeX[Rr]) + "\sin{\frac{" + (LaTeX[r]) + "t}{" + (LaTeX[R]) + "}}-" + (LaTeX[c]) + "\sin{\frac{" + (LaTeX[Rr]) + "t}{" + (LaTeX[R]) + "}}$, $t=" + (LaTeX[t]) + "$"