plotting - How to plot heatmap function over the unit circle

I have a function $Z(r, \theta)$ that I would like to plot over the unit disk. I could of course plot it as $Z(\sqrt{x^2+y^2}, \arctan(y/x))$, but the best I can do with this is a square plot. I would like a plot defined from $0

Zernike polynomial plots

How can I make Mathematica generate a plot like this? Thank you!

Answer

Here's my attempt to plot the Zernike functions on the unit disk:

ZernikeZ[n_Integer, m_Integer, r_, θ_] /; -n <= m <= n := 
         If[m < 0, Sin[m θ], Cos[m θ]] ZernikeR[n, m, r]

Table[DensityPlot[ZernikeZ[n, m, Norm[{x, y}], ArcTan[x, y]], {x, y} ∈ Disk[],
                  ColorFunction -> (ColorData[{"ThermometerColors",  "Reverse"},
                                              LogisticSigmoid[2 #]] &), 
                  ColorFunctionScaling -> False, Frame -> False, 
                  PlotPoints -> 55],
      {n, 0, 4}, {m, -n, n, 2}] // GraphicsGrid

Zernike on a disk

Comments

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