# Plot a curve in 2D [closed]

I have a curve $$\gamma=(f(x),g(x))$$ parameterized by a real parameter $$x > 1$$ (one can perform a variable change $$y = x^{-1} \in (0, 1)$$ if $$x\rightarrow\infty$$ is problematic), with the functions $$f(x)$$ and $$g(x)$$ known. I wish to plot this in 2D using Mathematica, but I don't know how to do it.

In my case $$$$f(x) = \frac{(1 - x^2)\cos(\theta)\sin(\theta)}{\cos^2(\theta) + \sin^2(\theta)x^2} \ , \quad g(x) = -\frac{x}{\cos^2(\theta) + \sin^2(\theta)x^2} \ ,$$$$

where $$\theta \in (0, \frac{\pi}{2}]$$ (I'm thinking of using Manipulate to plot different values of $$\theta$$).

So far I have only managed to use ListPlot to draw different points for $$x$$ in 2D:

Manipulate[ListPlot[Table[{f[x], g[x]}, {x, 1, 10}]], {\[Theta], 0, \[Pi]/2}]


I would like to draw the entire curve and not just some points along it. Help is appreciated.

• Have a look at Menu/WolframDocumentation/ParametricPlot. You can use Manipulate with it the same way you did in your attempt. – Alexei Boulbitch Mar 17 at 6:43

ParametricPlot[{((1 - x^2)*Cos[θ] Sin[θ])/(Cos[θ]^2 + Sin[θ]^2*x^2), -(x/(Cos[θ]^2 + Sin[θ]^2*x^2))}, {θ, 0, π/2}, {x, 1, 10}, MeshFunctions -> {#3 &, #4 &},
MeshStyle -> {Red, Cyan}, Mesh -> 20,
PlotStyle -> {Opacity[.1], Yellow}, PlotPoints -> 80]


Here is a solution with Manipulate:

Manipulate[
ParametricPlot[{((1 - x^2)*
Cos[\[Theta]] Sin[\[Theta]])/(Cos[\[Theta]]^2 +
Sin[\[Theta]]^2*
x^2), -(x/(Cos[\[Theta]]^2 + Sin[\[Theta]]^2*x^2))}, {x, 1,
50}, PlotRange -> {{-5, 0}, {-4, 0}}], {\[Theta], 0, \[Pi]/2}]



If we replace x -> 1/x we get:

Manipulate[
ParametricPlot[{((1 - x^2)*
Cos[\[Theta]] Sin[\[Theta]])/(Cos[\[Theta]]^2 +
Sin[\[Theta]]^2*
x^2), -(x/(Cos[\[Theta]]^2 + Sin[\[Theta]]^2*x^2))} /.
x -> 1/x, {x, 0, 1}, PlotRange -> {{0, 5}, {-4, 0}}], {\[Theta],
0, \[Pi]/2}]