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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.

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  • $\begingroup$ Does RevolutionPlot3D[{2 + Cos[t] + Sin[t], t, Sin[10 t]}, {t, 0, 2 Pi}] give what you need? $\endgroup$ – kglr Oct 1 '14 at 8:00
  • $\begingroup$ I don't think so. The plot should be a curve. $\endgroup$ – user14782 Oct 1 '14 at 8:01
  • $\begingroup$ ... or ParametricPlot3D[{2 + Cos[t] + Sin[t], t, Sin[10 t]}, {t, 0, 2 Pi}]? $\endgroup$ – kglr Oct 1 '14 at 8:04
  • 2
    $\begingroup$ 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. $\endgroup$ – user14782 Oct 1 '14 at 8:06
  • $\begingroup$ related: 67261 $\endgroup$ – Kuba Feb 11 '15 at 20:54
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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}]

enter image description here

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  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}]
    
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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}]

enter image description here

A nice curve it is. Have fun!

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  • 4
    $\begingroup$ 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}] $\endgroup$ – chuy Oct 1 '14 at 14:19

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