Change of coordinates and plots

I want to plot the following

$$L(\rho) = m + \frac{c}{\rho^2}$$

That's very straightforward.

y6[m_, c_, \[Rho]_] := m + c/\[Rho]^2
Plot[{y6[1, -2, r], y6[1, -0.5, r], y6[1, -0.1, r], y6[1, 0, r],
y6[1, 0.1, r], y6[1, 0.5, r], y6[1, 2, r]}, {r, 1, 10},
PlotLegends -> "Expressions", PlotRange -> {0.76, 1.25},
AxesOrigin -> {1.2, 0.84}]


The ($L$ vs $\rho$)-plot is

A valid change of coordinates is the following

$$\rho = r \sin(\theta)$$ $$L = r \cos(\theta)$$

And the corresponding plot is the $r \cos(\theta)$ vs $r \sin(\theta)$.

I don't know how to implement correctly the change of coordinates and how to write the command to get the $r \cos(\theta)$ vs $r \sin(\theta)$ plot.

Perhaps you are looking for this?

ParametricPlot[Evaluate@Table[{r Cos[θ], r Sin[θ]}, {r, 1, 10}],
{θ, 0, 2 Pi}, PlotLegends -> Array["r = " <> ToString[#] &, 10]]


Edit

Some overkill here, but out of time.

m = 1;
ParametricPlot[Evaluate@Flatten[
Table[{r Sin[θ], m + c/(r Sin[θ])^2}, {r, 1, 10},
{c, {-2, -0.5, -0.1, 0, 0.1, 0.5, 2}}], 1], {θ, 0, 2 Pi},
AxesLabel -> {"r Sin[θ]", "r Cos[θ] = m + c/(r Sin[θ])^2"},
ImageSize -> 550, BaseStyle -> 12,
PlotStyle -> Array[ColorData[97], 7],
PlotLegends -> Placed[Map["c = " <> ToString[#] &,
{-2, -0.5, -0.1, 0, 0.1, 0.5, 2}], Right],
AspectRatio -> 1/GoldenRatio]


• That is helpful indeed. Thank you. I am wondering if and how it is possible to include the specific relation between $r \sin u$ and $r \cos u$. That is using, $r \cos u = m + c/(r \sin u)^2$ and plot this particular relation with $r \cos u$ in the y-axis and $r \sin u$ in the x-axis. Feb 25 '18 at 14:12
• Thank you for the added bit. Great reply. Cheers!!! Feb 25 '18 at 14:58