https://www.perplexity.ai/search/5bfca7ae-ea6e-4cd6-b09d-480f4e88d6e0
Below are concrete parametric equations for all 30 variations. Use (t\in[0,2\pi k]), where (k) is chosen so the curve closes; for integer ratios (R/r), (k=r) is usually sufficient.
Base equations
For an epicycloid (rolling outside), with fixed-circle radius (R), rolling-circle radius (r), and tracing radius (d=r):
[ x(t)=(R+r)\cos t-d\cos!\left(\frac{R+r}{r}t\right), \qquad y(t)=(R+r)\sin t-d\sin!\left(\frac{R+r}{r}t\right). ]
For a hypocycloid (rolling inside):
[ x(t)=(R-r)\cos t+d\cos!\left(\frac{R-r}{r}t\right), \qquad y(t)=(R-r)\sin t-d\sin!\left(\frac{R-r}{r}t\right). ]
Setting (d\ne r) gives the broader epitrochoid and hypotrochoid families. journals.rudn
For compactness below, define:
[ E(R,r,d;t)=\left((R+r)\cos t-d\cos!\left(\frac{R+r}{r}t\right),\, (R+r)\sin t-d\sin!\left(\frac{R+r}{r}t\right)\right) ]
[ H(R,r,d;t)=\left((R-r)\cos t+d\cos!\left(\frac{R-r}{r}t\right),\, (R-r)\sin t-d\sin!\left(\frac{R-r}{r}t\right)\right) ]
where (E) is epitrochoidal and (H) is hypotrochoidal. A true cycloid has (d=r). journals.rudn
1–10: Classic curves
| # | Variation | Parametric equation | Parameters |
|---|---|---|---|
| 1 | Deltoid | (\gamma(t)=H(3,1,1;t)) | (R=3,\ r=1,\ d=1) |
| 2 | Astroid | (\gamma(t)=H(4,1,1;t)) | (R=4,\ r=1,\ d=1) |
| 3 | Five-petal epicycloid | (\gamma(t)=E(5,1,1;t)) | (R=5,\ r=1,\ d=1) |
| 4 | Six-lobed epicycloid | (\gamma(t)=E(6,1,1;t)) | (R=6,\ r=1,\ d=1) |
| 5 | Eight-cusp star | (\gamma(t)=H(8,1,1;t)) | (R=8,\ r=1,\ d=1) |
| 6 | Nephroid | (\gamma(t)=E(2,1,1;t)) | (R=2,\ r=1,\ d=1) |
| 7 | Cardioid | (\gamma(t)=E(1,1,1;t)) | (R=1,\ r=1,\ d=1) |
| 8 | Nine-cusp mandala | (\gamma_n(t)=H(9,1,1;t)) rotated by (n\cdot 2\pi/3) | Draw for (n=0,1,2) |
| 9 | Twelve-cusp flower | (\gamma(t)=H(12,1,1;t)) | (R=12,\ r=1,\ d=1) |
| 10 | Near-circle epicycloid | (\gamma(t)=E(30,1,1;t)) | (R=30,\ r=1,\ d=1) |
For #8, a rotation by angle (\alpha) is applied as
[
\gamma_\alpha(t)=
\begin{pmatrix}
\cos\alpha & -\sin\alpha
\sin\alpha & \cos\alpha
\end{pmatrix}
\gamma(t).
]
11–20: Tracing-point variations
| # | Variation | Parametric equation | Parameters |
|---|---|---|---|
| 11 | Inner-traced rosette | (\gamma(t)=H(7,3,0.55;t)) | (R=7,\ r=3,\ d=0.55) |
| 12 | Outer-traced rosette | (\gamma(t)=E(7,3,4.2;t)) | (R=7,\ r=3,\ d=4.2) |
| 13 | Progressive tracing radius | (\gamma_j(t)=H(7,3,d_j;t)), (d_j=0.2+0.13j) | (j=0,\dots,10) |
| 14 | Dual tracing points | (\gamma_\pm(t)=H(7,3,\pm 1;t)) | (d=1) and (d=-1) |
| 15 | Triple tracing points | (\gamma_m(t)=H(7,3,1;t)) rotated by (m\cdot 2\pi/3) | (m=0,1,2) |
| 16 | Offset tracing point | (\gamma(t)=H(7,3,1;t)+(0.35,0.18)) | Translation vector ((0.35,0.18)) |
| 17 | Elliptical tracing point | (\gamma(t)=H(7,3,1;t)+(0.35\cos 3t,0.18\sin 3t)) | Ellipse amplitudes (0.35,0.18); frequency (3) |
| 18 | Trailing point cloud | Sample (\gamma(t_i)=H(7,3,1;t_i)) | (t_i=2\pi i/N,\ i=0,\dots,N-1) |
| 19 | Speed-shaded curve | (\gamma(t)=H(7,3,1;t)), colour from (|\gamma’(t)|) | Shade by speed magnitude |
| 20 | Ghost trace | (\gamma_{\text{full}}(t)=H(7,3,1;t)), (\gamma_{\text{live}}(t)=H(7,3,1;t)) for (0\le t\le T) | Full curve faint; live segment bright |
For #19, the velocity is
[ \gamma’(t)= \left( -(R-r)\sin t -\frac{d(R-r)}{r}\sin!\left(\frac{R-r}{r}t\right), \; (R-r)\cos t -\frac{d(R-r)}{r}\cos!\left(\frac{R-r}{r}t\right) \right), ]
and the shading value can be (|\gamma’(t)|).
21–25: Combined families
| # | Variation | Parametric equation | Parameters |
|---|---|---|---|
| 21 | Inside–outside pair | (\gamma_1(t)=H(5,1,1;t)), (\gamma_2(t)=E(5,1,1;t)) | Same ratio (R/r=5) |
| 22 | Nested counter-rotation | (\gamma_{\text{outer}}(t)=H(9,2,1;t)), (\gamma_{\text{inner}}(t)=R_{-\pi/4}E(3,1,0.8;t)) | Outer (R=9,r=2); inner (R=3,r=1); inner rotated by (-\pi/4) |
| 23 | Phase-shifted twins | (\gamma_\pm(t)=H(7,3,1;t\pm\phi)) | (\phi=\pi/18) |
| 24 | Cycloidal gear profile | Addendum: (E(R_a,r_a,r_a;t)); dedendum: (H(R_h,r_h,r_h;t)), joined at the pitch circle | Choose compatible pitch radii; epicycloid forms the tooth tip, hypocycloid the root. matec-conferences |
| 25 | Hybrid flower | Outer: (E(6,1,1;t)); inner: (H(4,1,0.7;t)) | Scale inner curve by (0.45) before overlay |
For #24, a practical simplified profile is:
[
\gamma_{\text{tooth}}(t)=
\begin{cases}
E(R_a,r_a,r_a;t), & 0\le t\le \pi/2
H(R_h,r_h,r_h;t), & \pi/2<t\le\pi
\end{cases}
]
then repeat it (z) times around the gear centre with rotation (2\pi j/z).
26–30: Compositional and 3D forms
| # | Variation | Parametric equation | Parameters |
|---|---|---|---|
| 26 | Rotational symmetry field | (\gamma_j(t)=R_{2\pi j/n}H(7,3,1;t)) | (n=8), (j=0,\dots,n-1) |
| 27 | Spiral-scaled family | (\gamma_j(t)=s^jH(7,3,1;t)), (s=0.82) | (j=0,\dots,11) |
| 28 | Ribbon extrusion | (\Gamma(t,u)=\gamma(t)+u\,\mathbf n(t)w/2), (-1\le u\le1) | (\gamma(t)=H(7,3,1;t)), ribbon half-width (w/2) |
| 29 | Tube-swept sculpture | (\Gamma(t,v)=\gamma(t)+a\cos v\,\mathbf n(t)+a\sin v\,\mathbf b(t)) | (\gamma(t)=H(7,3,1;t)), tube radius (a) |
| 30 | Layered relief panel | (\Gamma_j(t)=\gamma(t)+j\Delta z\,\mathbf k), (\gamma(t)=H(7,3,1;t)) | Layers (j=0,\dots,4), depth spacing (\Delta z) |
For #28 and #29, use the Frenet-like frame of the planar curve:
[ \mathbf T(t)=\frac{\gamma’(t)}{|\gamma’(t)|}, \qquad \mathbf n(t)=\frac{(-T_y(t),T_x(t),0)}{|(-T_y(t),T_x(t),0)|}, \qquad \mathbf b(t)=(0,0,1). ]
Thus the 3D tube is
[
\Gamma(t,v)=
\begin{pmatrix}
\gamma_x(t)
\gamma_y(t)
0
\end{pmatrix}
+
a\cos v\,\mathbf n(t)
+
a\sin v\,\mathbf b(t),
\qquad
0\le v<2\pi.
]
Closing condition
For (R/r=p/q) in lowest terms, the curve closes after
[ t_{\max}=2\pi q. ]
For the integer-ratio examples above, (q=1), so (t\in[0,2\pi]) closes the curve. The hypotrochoid and epitrochoid formulas above are the standard generalized rolling-circle equations, with (d) controlling whether the result is a true cycloid or a curtailed/prolate trochoid. en.wikipedia