For a mapping $\gamma(x,y)$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ we can plot the image of some two dimensional region with ParametricPLot
as, for instance,
\[Gamma][x_, y_] := {x^2, x*y^3}
ParametricPlot[\[Gamma][x, y], Element[{x, y}, Disk[]], PlotRange -> All]
being the domain, say, a disk. But if I want to plot just the image of the boundary as
ParametricPlot[\[Gamma][x, y], Element[{x, y}, Circle[]], PlotRange -> All]
then ParametricPlot
has troubles understanding the dimensionality and the number of variables. Then, how can one make a parametric plot, being the domain some curve, as in the previous example ?
Note: I do not want to parametrize the circle as $(\cos\theta,\sin\theta)$ and then replace back to get $\gamma(\theta)$, because this may not be possible for more complicated curves without closed, analytical formula.
ParametricPlot[\[Gamma][x, y], Element[{x, y}, Disk[]], PlotRange -> All, BoundaryStyle -> Red, PlotStyle -> None]
$\endgroup$reg = ParametricRegion[{\[Gamma][x, y], {x, y} \[Element] Circle[]}, {x, y}]; Region[Style[reg, Red]]
$\endgroup$