Polar axes on a RegionPlot do not appear in correct location

Question

I'm trying to plot the region where some complicated function $f$ is positive, within the circular region 0<R<π/2. I based my code off this answer, however the resulting polar axes do not appear at π/2, as I specified.

eps = 1
RegionPlot[ Sin[5 x] + Cos[5 y] > 0.2 && Sqrt[x^2 + y^2] < \[Pi]/2,
 {x, -\[Pi]/2 - eps, \[Pi]/2 + eps}, {y, -\[Pi]/2 - eps, \[Pi]/2 + 
   eps},
 Frame -> False,
 Axes -> {False, True},
 Ticks -> {False, Range[0, \[Pi]/2, \[Pi]/8]},
 Epilog -> PolarPlot[{\[Pi]/4}, {\[Theta], 0, 2 \[Pi]},
    PolarAxes -> True,
    (*PlotRange\[Rule]\[Pi],*)
    PolarAxesOrigin -> {0, \[Pi]/2}, (* does not seem to do  what it's meant to *)
    PolarGridLines -> {Automatic, {0, \[Pi]/4, \[Pi]/2}},
    PolarTicks -> {Drop[Table[i, {i, 0, 2 Pi, Pi/6}], -1], None},
     PlotStyle -> {Black, Thin}
    ][[1]]]

Output:

Note that I'm deliberately not using the radial polar axis, since I can't find a simple way to make it plot in units of π.

Is this a mistake on my part, or an obscure bug?

Answer 1

It does look like "an obscure bug".

A work-around is to construct the polar axes/ticks primitives from scratch. The function polarAxes below creates angular and radial axes/ticks/gridlines given a radial range (radius), angular axis origin (startingangle) and specs for angular/radial ticks/gridlines (each can be specified as an integer or a list). The last two parameters control tick length and radial range padding, respectively.

ClearAll[polarAxes]
polarAxes[radius_: 1, startingangle_: Automatic, angulardivs_: Automatic, 
  radialdivs_: Automatic, angulargridlines_: Automatic, 
  radialgridlines_: Automatic, ticklength_: Automatic, axisoffset_: Automatic] := 
 Module[{angularticks, radialticks, 
   r = radius (1 + axisoffset /. Automatic -> .02), 
   sa = startingangle /. Automatic -> 0, tl = ticklength /. Automatic -> .05}, 
  angularticks = Switch[angulardivs, 
    _Integer, CirclePoints[{r, sa}, angulardivs],
    _List, r Transpose[Through@{Cos, Sin} @ Mod[angulardivs + sa, 2 π]],
    _, CirclePoints[{r, sa}, 12]]; 
  radialticks = Switch[radialdivs, 
    _Integer, Thread[{Subdivide[radius, radialdivs], 0}],
    _List, Thread[{radialdivs, 0}], 
    _, Thread[{Subdivide[radius, 4], 0}]]; 
  {GrayLevel[.4], AbsoluteThickness[.2],
   Circle[{0, 0}, r], 
   Line[{#, #  (1 + tl)}] & /@ angularticks,  
   Line[{{0, 0}, #}] & /@ Switch[angulargridlines, 
    _Integer, CirclePoints[{r, sa}, angulargridlines],
    _List, r Transpose[Through@{Cos, Sin}[Mod[angulargridlines + sa, 2 π]]], 
    _, angularticks],
   Circle[{0, 0}, #] & /@ Switch[radialgridlines, 
    _Integer, Subdivide[radius, radialgridlines], 
    _List, radialgridlines, 
    _, radialticks[[All, 1]]], 
   MapThread[Text[#, (1 + 3 tl/2) #2, -#2/2] &, 
     {Most @ Subdivide[0, 2 π, Length @ angularticks], angularticks}],
   GeometricTransformation[{
     MapThread[Rotate[Text[#, Offset[{0, -5}, #2 + {0, -r tl}], {0, 1}], -sa] &,
        {radialticks[[All, 1]], radialticks}], 
     Line[{#, # + {0, -r tl}}] & /@ radialticks}, 
    RotationTransform[sa]]}]

Examples:

eps = 1;
RegionPlot[Sin[5 x] + Cos[5 y] > 0.2 && Sqrt[x^2 + y^2] < π/2,
  {x, -π/2 - eps, π/2 + eps}, {y, -π/2 - eps, π/2 + eps}, 
 ImageSize -> 700, Frame -> False, Axes -> False, 
 Epilog -> {FontSize -> 12, polarAxes[π/2]}]

Use polarAxes[π/2, π/4] to make the angular axis start at π/4:

Use polarAxes[π/2, π/4, 8, Automatic, {0, π/6, π/4, π/2, π, 5 π/6, 17 π/12}, 5] to get 5 radial grid lines and angular grid lines at {0, π/6, π/4, π/2, π, 5 π/6, 17 π/12}:

Use polarAxes[π/2, π/4, {0, π/6, π/4, π/2, π, 5 π/6, 17 π/12}, 5, Automatic, 5] to get angular ticks and gridlines at specified angles and 5 radial ticks/gridlines:

Answer 2

Consider:

PolarTicks -> {Drop[Table[i, {i, 0, 2 Pi, Pi/6}], -1], None}

This specifies that you do not want an r-axis with radial ticks. Change "None" e.g. to "Automatic":

eps = 1
RegionPlot[
 Sin[5 x] + Cos[5 y] > 0.2 && 
  Sqrt[x^2 + y^2] < \[Pi]/2, {x, -\[Pi]/2 - eps, \[Pi]/2 + 
   eps}, {y, -\[Pi]/2 - eps, \[Pi]/2 + eps}, Frame -> False, 
 Axes -> {False, True}, Ticks -> {False, Range[0, \[Pi]/2, \[Pi]/8]}, 
 Epilog -> 
  PolarPlot[{\[Pi]/4}, {\[Theta], 0, 2 \[Pi]}, 
    PolarAxes -> True,(*PlotRange\[Rule]\[Pi],*)
    PolarAxesOrigin -> {0, \[Pi]/2}, 
    PolarGridLines -> {Automatic, {0, \[Pi]/4, \[Pi]/2}}, 
    PolarTicks -> {Drop[Table[i, {i, 0, 2 Pi, Pi/6}], -1], Automatic},
 PlotStyle -> {Black, Thin}][[1]]]

Or with rotated r-axis:

eps = 1
RegionPlot[
 Sin[5 x] + Cos[5 y] > 0.2 && 
  Sqrt[x^2 + y^2] < \[Pi]/2, {x, -\[Pi]/2 - eps, \[Pi]/2 + 
   eps}, {y, -\[Pi]/2 - eps, \[Pi]/2 + eps}, Frame -> False, 
 Axes -> {False, True}, Ticks -> {False, Range[0, \[Pi]/2, \[Pi]/8]}, 
 Epilog -> 
  PolarPlot[{\[Pi]/4}, {\[Theta], 0, 2 \[Pi]}, 
    PolarAxes -> True,(*PlotRange\[Rule]\[Pi],*)
    PolarAxesOrigin -> {Pi/4, Pi/2}, 
    PolarGridLines -> {Automatic, {0, \[Pi]/4, \[Pi]/2}}, 
    PolarTicks -> {Drop[Table[i, {i, 0, 2 Pi, Pi/6}], -1], Automatic},
     PlotStyle -> {Black, Thin}][[1]]]