How do I rescale tick marks on a polar plot?

Question

I am working with an external "number-crunching" application that generates radiation patterns for antennas. The data that I am working with is a function of spherical angles theta and phi and it produces numbers in the range of 0 through some maximum value. I want to plot using PolarPlot to show a "scaled" tick mark where the scaling is the result of some function that I write to do the proper transformation. So, here is an example [below, phi is controlled externally and sometimes via Manipulate[]).

PolarPlot[
  gdb[180/Pi (Abs[Pi/2 - theta]), phi], {theta, 0, Pi}, 
  PolarAxes -> True, PolarTicks -> {"Degrees", Automatic}, 
  PolarGridLines -> Automatic, PlotRangeClipping -> False, 
  PolarAxesOrigin -> Automatic]

And, this is the resultant plot:

Now, the PolarAxesOrigin option given Automatic generates the axes labeled from 20 thru the maximum of 80. However, for this same plot, the only change I want (in this example) is to scale that axes tick mark labels as 0.0 for the 80 tick mark label and then the inner concentric circles would be -10, -20, -30, -40, and so on. Thus, the labeling are all negative numbers with zero as the maximum range on the PolarAxesOrigin.

I have researched various posted questions here and experimented using scaling features, tick functions, and other things (merely guessing) to figure out how to achieve this and nothing works right. I am hoping that I can provide some kind of function that can do a transformation from the plotted values to some scaled tick mark labels. Changing the data itself to be negative does not work, the resultant plot is not correct.

Answer 1

It's a shame you did not include your own function, so I will have to use a made up one. I do not know how to change that detail before the plot is generated, but inspection of the plot's InputForm (try plot // InputForm // SequenceForm) shows the general format of those labels. This can be modified after the fact as follows, using a global replacement to generate the ticks you want:

plot = PolarPlot[
         25 theta, {theta, 0, Pi},
         PolarAxes -> True,
         PolarTicks -> {"Degrees", Automatic},
         PolarGridLines -> Automatic,
         PlotRangeClipping -> False,
         PolarAxesOrigin -> Automatic
       ];

plot /. Text[Style[value_Real, {}], opts__] :> Text[Style[value - 80, {}], opts]

Above I restricted the pattern for replacement using val_Real to avoid causing the same change to the angular ticks, which otherwise would also become e.g. 30° - 80. "30°" is represented internally as Times[30, Degree]; that expression has head Times to Mathematica's eyes, and is not a real number (i.e. something with head Real), so it can be distinguished from the radial ticks that are plain real numbers instead.

Answer 2

Yes, for some reason Ticks does not work with PolarPlot. I would prefer a more straight forward solution because of the obvious side effect of the above hack. Additionally its more versatile to produce ticks and other items.

pplot = PolarPlot[ 
    { Style[ 4, Thick, Gray ], 
      Style[ 2 Sin[3x] + 4, Blue, Thick ] }, { x, 0, 2\[Pi] }, 
PlotRangeClipping -> False,
PolarAxes -> { True, False },
PolarGridLines -> { Range[12] * 30 Degree, Range[10] },
PolarTicks -> { "Degrees", Automatic },
PolarAxesOrigin -> { 0, 10 },
Epilog -> Table[Table[ Text[ v-4, pos ], { v, Drop[Range[9],{4}] } ], 
    { pos, { {0,v}, {0,-v}, {v,0}, {-v,0} } } ]
];

The radial axis is swiched off and substituted in the Epilog part.

Answer 3

PolarTicks >> Details:

• For each axis, the following tick mark options can be given:

So we can use

radialTicks = Thread[{Subdivide[0, 80, 8], 
   Style[-#, 16, Red] & /@ Reverse[Subdivide[0, 80, 8]], .01, Green}]

PolarPlot[20 (2 + Sin[3 t]), {t, 0, 2 Pi}, ImageSize -> Large, 
 PolarAxes -> True, 
 PolarGridLines -> {Subdivide[##, 24] &, Range[##, 10] &}, 
 PolarTicks -> {"Degrees", radialTicks}, 
 PolarAxesOrigin -> {Pi/2, 80}]

Alternatively, we can use a ticks function that uses the automatic radial axis range to construct a list of labeled ticks:

radialTicksFunction = Thread[{Subdivide[##, 8], 
    Style[-#, 16, Red] & /@ Reverse[Subdivide[##, 8]], .01, Green}] &;

PolarPlot[20 (2 + Sin[3 t]), {t, 0, 2 Pi}, ImageSize -> Large, 
 PolarAxes -> True, 
 PolarGridLines -> {Subdivide[##, 24] &, Range[##, 10] &}, 
 PolarTicks -> {"Degrees", radialTicksFunction}, 
 PolarAxesOrigin -> {Pi/2, 80}]

to get