Angular Range in Polar Plot

Question

I am trying to make a polar plot using the following code

PolarPlot[1, {θ, 0, Pi/3}, PolarAxes -> True, 
 PolarTicks -> {"Degrees", Automatic}, PolarGridLines -> True, 
 PlotRange -> 1.5]

What I would like to do is to set the angle range of the polar axis, from 0 to 60 degrees. I've tried using PlotRange but this changes the radial component of the function...

How is it possible to define an anglular range of a polar plot?

Answer 1

Edit

Original post at the end. This is uglier, but cleaner and more robust:

tmax = Pi/3;
rmax = 1.5;
u = PolarPlot[t (*Your function Here*), {t, 0, Pi}, 
   PolarAxes -> {True, True}, PolarTicks -> {"Degrees", Automatic}, 
   PolarGridLines -> True, PlotRange -> rmax, 
   RegionFunction -> Function[{x, y, t, r}, t < tmax], 
   PolarAxesOrigin -> {0, rmax}, PolarAxes -> 0];
Show[Quiet@
 Replace[u,
   {Circle[{0, 0}, x_, {0, 2 Pi}] -> Circle[{0, 0}, x, {0, tmax}], 
    Line[{x_, y_}] /;  tmax < ArcTan[y[[1]], y[[2]]] || 0 > ArcTan[y[[1]], y[[2]]] :> {},
    Line[{Scaled[x1_, y1_], Scaled[x2_, y2_]}] /; 
                                         tmax < ArcTan[y2[[1]] + x2[[1]], y2[[2]] + x2[[2]]] || 
                                         0 > ArcTan[y2[[1]] + x2[[1]], y2[[2]] + x2[[2]]] :> {},
    {{a_ (Sin | Cos)[y_], b_  (Sin | Cos)[y_]}, 
                                   Scaled[{s_, t_}, {c_ (Sin | Cos)[y_], d_ (Sin | Cos)[y_]}]} /; 
                                                   ArcTan[s, t] > tmax || ArcTan[s, t] < 0 -> {},
    Text[Style[TraditionalForm[Times[x_, Degree]], List[]], __] /; x > tmax (180/Pi) :> {}}, 
    Infinity], 
 PlotRange -> {{0, rmax}, {0, rmax Sin[tmax]}}]

Original post I know this is no beauty, but just an idea:

rmax = 1.5;
Show[
  PolarPlot[1.3 t (*Your function Here*), {t, 0, Pi}, 
            PolarAxes -> {True, False},  
            PolarTicks -> {"Degrees", Automatic}, PolarGridLines -> True, 
            PlotRange -> rmax, 
            RegionFunction -> Function[{x, y, t, r}, t < Pi/3],
            PolarAxesOrigin -> {0, rmax}],

  Graphics[{White, Disk[{0, 0}, rmax  2, {0 - 1/15, -5/3 Pi + 1/20}]}],

  PolarPlot[rmax, {t, 0, Pi/3}, PolarAxes -> {False, True}, 
            PolarTicks -> {None, Automatic}, PolarGridLines -> False, 
            PlotRange -> rmax, PolarAxesOrigin -> {0, rmax},
            PolarAxes -> 0]
 ,PlotRange -> {{0, rmax}, {0, rmax Sin[Pi/3]}}]

The

{0 - 1/15, -5/3 Pi + 1/20}

needs some elaboration, for sure.

Answer 2

What is most interesting is, unlike Plot, PolarPlot uses explicit graphics primitives to present axes and grids, so we can "filter" out things which is out of our interested range by force (though which is not a beautiful method).

Here is my brute-force function. I've made it capable of dealing with range crossing $\theta=0$, but yet added a feature which should rotate the axes into the interested range when it's not.

Clear[polarRangeFunc]
polarRangeFunc[graph_, \[Alpha]_, \[Beta]_] := 
 Module[{\[Theta]min, \[Theta]max, sgn, rangeFunc},
  {\[Theta]min, \[Theta]max} = Sort[Mod[{\[Alpha], \[Beta]}, 2 \[Pi]]];
  sgn = Sign[\[Alpha] \[Beta]];
  rangeFunc[pos_] := 
   If[pos == {0, 0}, False, 
    If[sgn == -1, #, Not@#] &[\[Theta]min <= 
      Mod[Arg[{1, I}.pos], 2 \[Pi]] <= \[Theta]max]];
  graph /. (PlotRange -> _) :> AbsoluteOptions[Graphics[{
          Disk[{0, 0}, 
           Max[Abs[
             PlotRange /. 
              AbsoluteOptions[graph, PlotRange]]], {\[Alpha], \[Beta]}]
          }], PlotRange] //
     ReplacePart[#, {1, 5} -> (#[[1, 5]] /.
          {Line[{Scaled[_, pos_], Scaled[_, pos_]}] /; 
             rangeFunc[pos] :> Line[{}],
           annoymousTicks_?(MatchQ[#, 
                Line[{{{_, _}, Scaled[__]} ..}]] &) :>
            (annoymousTicks /. {pos_, Scaled[__]} /; rangeFunc[pos] :>
                Sequence[]),
           Text[_, 
              Offset[_, pos_], ___] /; (rangeFunc@
               If[Head[pos] === Scaled, pos[[2]], pos]) :> Sequence[],
           Circle[orig_, radius_, angleRange_] :> 
            Circle[orig, radius, {\[Alpha], \[Beta]}]}
         )] & //
    ReplacePart[#, {1, 1} -> (#[[1, 1]] /.
         {Line[{{0, 0}, pos_}] /; rangeFunc[pos] :> Line[{}],
          Circle[orig_, radius_, angleRange_] :> 
           Circle[orig, radius, {\[Alpha], \[Beta]}]}
        )] & //
   ReplacePart[#, {1, 3} -> (#[[1, 3]] /.
        Line[pts_] :> 
         Line[pts /. {x_, y_} /; rangeFunc[{x, y}] :> "outpt" //
           (SplitBy[#, StringQ] /. "outpt" -> Sequence[]) &]
       )] &
  ]

And here is an example:

polorgraph = 
 PolarPlot[{2 \[Theta]^(-1/3), 3 Cos[\[Theta]^(1/2)]},
 {\[Theta], .01, 10 \[Pi]}, 
  PlotStyle -> {Directive[Red, Thick], Directive[Purple, Thick]},
  PolarAxes -> True, PolarAxesOrigin -> {\[Pi]/6, 3}, PlotRange -> 3, 
  PolarTicks -> {"Degrees", Automatic}, PolarGridLines -> True]

polarRangeFunc[polorgraph, \[Pi]/6, \[Pi]/2]

polarRangeFunc[polorgraph, -\[Pi]/6, 6 \[Pi]/5]