# Plotting conformal mappings: $(x, y) \mapsto (x^2, y)$

I'm currently studying transformation geometry, and I've been trying to figure out a good way to plot conformal mappings on $$\mathbb{R}^2$$ of the form $$\mathbf{f} : \pmatrix{x \\ y} \mapsto \pmatrix{f_1(x) \\ f_2(y)} ,$$ for example with $$f_1(x) = x^2$$ and $$f_2(y) = y$$. All the examples I've seen so far (e.g. here and here) are in the complex plane, with a single valued function, and seem to be complex analysis stuff.

For context, I was inspired by the animations in 3Blue1Brown's videos. However, in addition to linear transformations like $$R = \pmatrix{ \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) } ,$$ I'm also interested in seeing the effect of nonlinear functions on the gridlines of the Cartesian plane.

Also from complex analysis, just by using Re and Im to obtain f1[x,y] and f2[x,y]

    f1[x_, y_] = x^2 - y^2;
f2[x_, y_] = 2 x*y;
ParametricPlot[{f1[x, y], f2[x, y]}, {x, -4, 4}, {y, -4, 4},
MeshFunctions -> Automatic, Mesh -> 8,
MeshShading -> {{LightRed, LightGreen}, {LightBlue, LightYellow}},
Axes -> False, PlotRange -> All, BoundaryStyle -> None,
FrameLabel -> {u, v}, LabelStyle -> {FontFamily -> "Times", Blue},
PlotPoints -> 50]


Black and White

f1[x_, y_] = x^2 - y^2;
f2[x_, y_] = 2 x*y;
ParametricPlot[{f1[x, y], f2[x, y]}, {x, -4, 4}, {y, -4, 4},
MeshFunctions -> Automatic, Mesh -> 8,
MeshStyle -> Directive[Thickness[0.015], Cyan],
MeshShading -> {{Black, White}, {White, Black}}, Axes -> False,
PlotRange -> All, BoundaryStyle -> Red, FrameLabel -> {u, v},
LabelStyle -> {FontFamily -> "Times", Blue}, PlotPoints -> 50] • That's perfect. Is there a way to overlay the mapping on the original (ideally black and white) grid? – Jeremy Lindsay Sep 4 at 8:35
• What I meant was to have the deformed grid on top of the original grid. Thanks! – Jeremy Lindsay Sep 4 at 8:47