What equation will create a 3D rose curve?

Question

The parametric equation $x=a\cos(bt)\cos(t)$, $y=a\cos(bt)\sin(t)$ where $a$ & $b$ are constants and $t$ is parameter gives a rose curve which looks like,

On a similar basis, is there a equation that gives a 3D rose curve? The curve would look like the surface formed by rotating each of the "petal" of the rose curve in 360 degrees along the radius vector.(I hope you get what I want to say -;)

Answer 1

I made a video on this. Are you the one person that seen it and hit the like button on it?

https://www.youtube.com/watch?v=Y7utC53CNs4

I think these are nd rose curves. Assume x_value is from the spherical coordinates on the wikipedia page

http://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates r=x_1+x_2+..x_n

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For 3d

$x_1=cos(\phi_1)*(x_1+x_2+x_3)$

$x_2=sin(\phi_1)*cos(\phi_2)*(x_1+x_2+x_3)$

$x_3=sin(\phi_1)*sin(\phi_2)*(x_1+x_2+x_3)$

Answer 2

This doesn't exactly answer the question, but with $k$, $m$, and $n$ positive integers, the parametric equations \begin{alignat*}{3} x(s, t) &= a\cos(mt) \cos^{k}(ns) &&\cos(t) &&\cos(s), \\ y(s, t) &= a\cos(mt) \cos^{k}(ns) &&\sin(t) &&\cos(s), \\ z(s, t) &= a\cos(mt) \cos^{k}(ns) &&\sin(s) && \end{alignat*} may provide some enjoyable plotting along similar lines.

For example, here's the surface with $m = 4$, $n = 1$, and $k = 8$:

The underlying idea is to take $\rho = \cos(m\theta)\cos^{k}(n\phi)$ in spherical coordinates $$ (x, y, z) = (\rho\cos\theta \cos\phi, \rho\sin\theta \cos\phi, \rho\sin\phi). $$

You may also enjoy learning about spherical harmonics.