# Spherical parametric plot

How can I plot such a parametrized curve:

$r = 2\cos t + \sin t$

$\theta = t$

$\phi = \sin 10t$

where $0<t<\pi/2$.

In the documentation only has been discussed parametric plot in Cartesian coordinates.

• Does RevolutionPlot3D[{2 + Cos[t] + Sin[t], t, Sin[10 t]}, {t, 0, 2 Pi}] give what you need?
– kglr
Commented Oct 1, 2014 at 8:00
• I don't think so. The plot should be a curve.
– user14782
Commented Oct 1, 2014 at 8:01
• ... or ParametricPlot3D[{2 + Cos[t] + Sin[t], t, Sin[10 t]}, {t, 0, 2 Pi}]?
– kglr
Commented Oct 1, 2014 at 8:04
• I found in documentation that ParametricPlot3D treats the first three slots as x, y and z variables. But my problem is parametrized r, theta and phi.
– user14782
Commented Oct 1, 2014 at 8:06
• related: 67261
– Kuba
Commented Feb 11, 2015 at 20:54

If you have version 9+ you can use:

ParametricPlot3D[
Evaluate[CoordinateTransform["Spherical" -> "Cartesian",{2 Cos[t]+Sin[t],t,Sin[10 t]}]],
{t,0,Pi/2}]

1. covert your spherical parametrized curve into cartesian parametrized curve with formulas:

$$\begin{cases} x&=r\sin\theta\cos\varphi\\ y&=r\sin\theta\sin\varphi\\ z&=r\cos\theta \end{cases}$$

in Mathematica better use some function for this:

ConvertToCartesianParametr[r_, theta_,phi_] :=
{r*Sin[theta]*Cos[phi], *Sin[theta]*Sin[phi], r*Cos[theta]}
1. use ParametricPlot3D as mentioned by Alexei Boulbitch

r = 2*Cos[t] + Sin[t];
theta = t;
phi = Sin[10*t];
ParametricPlot3D[ConvertToCartesianParametr[r, theta, phi], {t, 0, 2*Pi}]

It is just the answer of @kguler, with a slight modification. Try this:

ParametricPlot3D[{(2 + Cos[t] + Sin[t])*Sin[t]*
Cos[Sin[10 t]], (2 + Cos[t] + Sin[t])*Sin[t]*
Sin[Sin[10 t]], (2 + Cos[t] + Sin[t])*Cos[t]}, {t, 0, 2 Pi}]

A nice curve it is. Have fun!

• If you have version 9+ you can use: ParametricPlot3D[Evaluate[CoordinateTransform[ "Spherical" -> "Cartesian",{2 Cos[t]+Sin[t],t,Sin[10 t]}]],{t,0,Pi/2}]
– chuy
Commented Oct 1, 2014 at 14:19