cykloida

$\varphi=%v$

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

K

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

φ

φ

$\text{Angle }\varphi$

$\text{Circle }K$

$\text{Circle }k$

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

$\text{Point }B$

$\text{Hypo/Pericycloid}$

R = 3 (slider from 0 to (10 * r), step 0.1)
φ = 6.1 (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(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)
i = Circle((0, 0), R)
d = GCD((10 * R), (10 * r))
j = Segment((0, 0), S)
e = Max({R - r + c, R, r - R + c, R + r + c, (2 * r) - R})
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)"]]]) + ""
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))
g = Ray((0, 0), S)
A = Intersect(i, g)
α = Angle((R, 0), (0, 0), A)
h = Ray(S, (0, 0))
C, D = Intersect(i, h)
β = Angle((R, 0), (0, 0), D)
m = Segment(D, (0, 0))
l = true
n = true
o = true
p = true
q = true
s = true
Rr: R - r
text1_1: "$\textstyle x=" + (LaTeX[Rr]) + "\cos{\varphi}+" + (LaTeX[c]) + "\cos{\frac{" + (LaTeX[Rr]) + "\,\varphi}{" + (LaTeX[r]) + "}}$, $\textstyle y=" + (LaTeX[Rr]) + "\sin{\varphi}-" + (LaTeX[c]) + "\sin{\frac{" + (LaTeX[Rr]) + "\,\varphi}{" + (LaTeX[r]) + "}}$, $\varphi=" + (LaTeX[φ]) + "$"