# What equation will create a 3D rose curve?

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 -;)

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.

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

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

For 3d

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

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

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

• Can you put the equation for the curve here? – Registered User Mar 30 '15 at 10:38
• n-sphere multiplied by the sum of the n-sphere on all axis which ends up being a scalar. It becomes the radius. Same thing can be applied to many 3d objects like the sphere. – user1698948 Mar 31 '15 at 10:51
• Sorry, that general equation is wrong. Glad I did it wrong here and realized it before anyone noticed. For a website filled with know it alls that do everything in their power to belittle people I'm surprised the glaring mistake went unnoticed for 2 days. – user1698948 Apr 2 '15 at 12:47
• @user1698948 — I wanted to see your youtube video, but it's not working... Is there any other way I could view it? – space Nov 28 '18 at 2:33