cykloida

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

K

t

\text{Change }r<R\text{ to Hypocycloid or }r>R\text{ to Pericycloid
}

t

$\text{Angle }t$

$\text{Circle }K$

$\text{Circle }k$

$\text{Hypo/Pericycloid [shadow]}$

$\text{Point }B$

$\text{Hypo/Pericycloid}$

R = 3 (slider from 0 to (10 * r), step 0.1)
t = 17.8 (slider from 0 to ((20 * R / d) * π), step 0.1)
r = 1 (slider from 0.1 to 10, 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)
d = GCD((10 * R), (10 * r))
j = Segment((0, 0), S)
A = Intersect(k, h)
g = Segment(A, S)
e = Max({R - r + c, R, r - R + c, R + r + c, (2 * r) - 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)
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))
l = Ray((0, 0), S)
C = Intersect(i, l)
α = Angle(A, S, C)
m = Segment(S, C)
n = Ray(S, (0, 0))
D = Intersect(k, n)
β = Angle(B, S, D)
text3: "Pericycloid" + (LaTeX[If[r ≟ c, "\text{ Ordinary }(c=r)", If[r  <  c, "\text{ Curtate }(c>r)", "\text{ Prolate }(c<r)"]]]) + ""
text2: "Hypocycloid " + (LaTeX[If[r ≟ c, "\text{ Ordinary }(c=r)", If[r > c, "\text{ Curtate }(c<r)", "\text{ Prolate }(c>r)"]]]) + ""
o = true
p = true
q = true
s = true
u = true
v = true
Rr: R - r
text1_1: "$\textstyle x=" + (LaTeX[Rr]) + "\cos{\frac{" + (LaTeX[r]) + "t}{" + (LaTeX[R]) + "}}+c\cos{\frac{" + (LaTeX[Rr]) + "t}{" + (LaTeX[R]) + "}}$, $\textstyle y=" + (LaTeX[Rr]) + "\sin{\frac{" + (LaTeX[r]) + "t}{" + (LaTeX[R]) + "}}-c\sin{\frac{" + (LaTeX[Rr]) + "t}{" + (LaTeX[R]) + "}}$, $t=" + (LaTeX[t]) + "$"