Scalloped edges to Plot3D with Disc domain

Question

I'm running into problems with a weird edges to my Plot3D with a Disc domain. MWE:

Plot3D[(x^2 + y^2 - 100)^2, {x, y} \[Element] Disk[{0, 0}, 13.5], PlotPoints -> 100]

which produces

I don't know where the scalloping is coming from. It disappears for low PlotPoints, e.g. 10, but I need high plotpoints for the colour scheme I'm using. I've tried using Show[] with a slightly lower PlotRange to try to clip them off, but they remain.

I would dearly love a version with a crisp straight edge for my paper - would be grateful for any fixes!

Answer 1

Plot3D[(x^2 + y^2 - 100)^2, {x, -14, 14}, {y, -14, 14}, 
 PlotPoints -> 100, 
 RegionFunction -> 
  Function[{x, y}, {x, y} ∈ Disk[{0, 0}, 13.5]]]

Or

rm = RegionMember[Disk[{0, 0}, 13.5]]; 
Plot3D[(x^2 + y^2 - 100)^2, {x, -14, 
  14}, {y, -14, 14}, PlotPoints -> 100, 
 RegionFunction -> Function[{x, y}, rm@{x, y}]]

Answer 2

1. Rather than manually increasing the number of PlotPoints, increase the MaxRecursion parameter instead, to let the built-in plotting machinery decide where to use extra points (e.g. in areas of high curvature). This can often be much more efficient. The value of MaxRecursion -> 4 shown below was found by trial and error, starting low and increasing it until a satisfactory result was achieved.

2. As @cvgmt showed in their answer as well, use a pre-calculated RegionMemberFunction obtained from RegionMember as your RegionFunction in Plot3D.

This results in the following code:

rmf = RegionMember[Disk[{0, 0}, 13.5]];

Plot3D[
  (x^2 + y^2 - 100)^2, {x, -14, 14}, {y, -14, 14},
  RegionFunction -> (rmf[{#1, #2}] &),
  MaxRecursion -> 4
]

This approach is quite a bit faster than the alternative with explicit PlotPoints: on my computer this version takes roughly 0.5 s, whereas the equivalent approach with PlotPoints -> 100 takes roughly 1.5 s.