Why does Dynamic for PlotRange in ListLinePlot prevent the proper range from being used?

Question

I am using Manipulate to simulate Brownian motion. I want to be able to zoom in. Thus, I want to use Dynamic for the PlotRange option so that this can be done quickly without recomputing the motion. It works properly without Dynamic (except for speed), but using Dynamic changes its operation so that the range is no longer what I ask for once I zoom in sufficiently. See this by changing the "horizontal zoom". I also have the same PlotRange in Show. It seems I need both. No doubt I am not doing other things optimally, but this issue is a separate problem. Here is my code, first without the first Dynamic and second with it. At the end is a smaller example. It behaves differently, but again the behavior depends on whether PlotRange has Dynamic or not. This last example contains both types of code in it. It comes in two forms, one using Show and one not.

Manipulate[
Module[{bmPath, iterate}, SeedRandom[seed]; 
iterate[l_List] := 
Module[{i = 1, len = Length[l], new = l}, 
While[i < Length[new], 
 new = Insert[
   new, (new[[i]] + new[[i + 1]])/2 + 
    RandomVariate[NormalDistribution[0, .5/Sqrt[(len - 1)]]], 
   i + 1]; i = i + 2]; new]; 
bmPath = {0, RandomVariate[NormalDistribution[0, 1]]}; 
Do[bmPath = iterate[bmPath], {n}]; 
Show[ListLinePlot[bmPath, AxesLabel -> {"Time", "Location"}, 
DataRange -> {0, 1}, 
PlotRange -> {Clip[{hcenter - 2^(-zoom), hcenter + 2^(-zoom)}, {0,
     1}], vcenter + {-2, 2}/2^(scale/2)}], 
Plot[{Sqrt[x], -Sqrt[x]}, {x, 0, 1}, 
PlotStyle -> If[bound, Dotted, Opacity[0]]], 
PlotRange -> 
Dynamic[{Clip[{hcenter - 2^(-zoom), hcenter + 2^(-zoom)}, {0, 1}],
   vcenter + {-2, 2}/2^(scale/2)}], 
AxesOrigin -> Dynamic[{hcenter, vcenter}], 
ImageSize -> {475, 360}]],
{{n, 10, "detail"}, 1, 40, 1, Appearance -> {"Labeled", "Open"}, 
AnimationRate -> rate}, {{rate, 1, "rate"}, 1, 3, 1, 
Appearance -> "Labeled"}, {{bound, False, 
"show sqrt bounds"}, {True, False}}, Delimiter, {{hcenter, .5, 
"horizontal center"}, 0, 1, .01/2^(zoom - 1), 
Appearance -> {"Labeled", "Open"}, 
AnimationRate -> .2/2^(zoom - 1)}, {{vcenter, 0, 
"vertical center"}, -3, 3, .01/2^(scale - 1), 
Appearance -> {"Labeled", "Open"}, 
AnimationRate -> .1/2^(scale - 1)}, Delimiter, {{zoom, 1, 
"horizontal zoom"}, 1, 20, 1, 
Appearance -> {"Labeled", "Open"}}, {{scale, 1, 
"vertical scale"}, -2, 20, 1, 
Appearance -> {"Labeled", "Open"}}, Delimiter, {{seed, 77777, 
"new random case"}, 10000, 999999, 1, Appearance -> "Open", 
AnimationRate -> newrate}, {{newrate, 1, "rate of new case"}, 1, 10,
1, Appearance -> "Labeled"},
ControlPlacement -> Right, SynchronousUpdating -> False]


Manipulate[
Module[{bmPath, iterate}, SeedRandom[seed]; 
iterate[l_List] := 
Module[{i = 1, len = Length[l], new = l}, 
While[i < Length[new], 
 new = Insert[
   new, (new[[i]] + new[[i + 1]])/2 + 
    RandomVariate[NormalDistribution[0, .5/Sqrt[(len - 1)]]], 
   i + 1]; i = i + 2]; new]; 
bmPath = {0, RandomVariate[NormalDistribution[0, 1]]}; 
Do[bmPath = iterate[bmPath], {n}]; 
Show[ListLinePlot[bmPath, AxesLabel -> {"Time", "Location"}, 
DataRange -> {0, 1}, 
PlotRange -> Dynamic[{Clip[{hcenter - 2^(-zoom), hcenter + 2^(-zoom)}, {0,
     1}], vcenter + {-2, 2}/2^(scale/2)}]], 
Plot[{Sqrt[x], -Sqrt[x]}, {x, 0, 1}, 
PlotStyle -> If[bound, Dotted, Opacity[0]]], 
PlotRange -> 
Dynamic[{Clip[{hcenter - 2^(-zoom), hcenter + 2^(-zoom)}, {0, 1}],
   vcenter + {-2, 2}/2^(scale/2)}], 
AxesOrigin -> Dynamic[{hcenter, vcenter}], 
ImageSize -> {475, 360}]],
{{n, 10, "detail"}, 1, 40, 1, Appearance -> {"Labeled", "Open"}, 
AnimationRate -> rate}, {{rate, 1, "rate"}, 1, 3, 1, 
Appearance -> "Labeled"}, {{bound, False, 
"show sqrt bounds"}, {True, False}}, Delimiter, {{hcenter, .5, 
"horizontal center"}, 0, 1, .01/2^(zoom - 1), 
Appearance -> {"Labeled", "Open"}, 
AnimationRate -> .2/2^(zoom - 1)}, {{vcenter, 0, 
"vertical center"}, -3, 3, .01/2^(scale - 1), 
Appearance -> {"Labeled", "Open"}, 
AnimationRate -> .1/2^(scale - 1)}, Delimiter, {{zoom, 1, 
"horizontal zoom"}, 1, 20, 1, 
Appearance -> {"Labeled", "Open"}}, {{scale, 1, 
"vertical scale"}, -2, 20, 1, 
Appearance -> {"Labeled", "Open"}}, Delimiter, {{seed, 77777, 
"new random case"}, 10000, 999999, 1, Appearance -> "Open", 
AnimationRate -> newrate}, {{newrate, 1, "rate of new case"}, 1, 10,
1, Appearance -> "Labeled"},
ControlPlacement -> Right, SynchronousUpdating -> False]


Manipulate[
Module[{bmPath}, bmPath = {0, 2, 1}; 
Row[{ListLinePlot[bmPath, AxesLabel -> {"Time", "Location"}, 
 DataRange -> {0, 1}, 
 PlotRange -> 
  Dynamic[{Clip[{.5 - 2^(-zoom), .5 + 2^(-zoom)}, {0, 1}], {-2, 
     2}}], ImageSize -> {200, 100}], 
ListLinePlot[bmPath, AxesLabel -> {"Time", "Location"}, 
 DataRange -> {0, 1}, 
 PlotRange -> {Clip[{.5 - 2^(-zoom), .5 + 2^(-zoom)}, {0, 
     1}], {-2, 2}}, ImageSize -> {200, 100}]}]],
{{zoom, 1, "horizontal zoom"}, 1, 20, 1, 
Appearance -> {"Labeled", "Open"}},
ControlPlacement -> Right, SynchronousUpdating -> True]


Manipulate[
Module[{bmPath}, bmPath = {0, 2, 1}; 
Row[{ListLinePlot[bmPath, AxesLabel -> {"Time", "Location"}, 
 DataRange -> {0, 1}, 
 PlotRange -> 
  Dynamic[{Clip[{.5 - 2^(-zoom), .5 + 2^(-zoom)}, {0, 1}], {-2, 
     2}}], ImageSize -> {200, 100}], 
Show[ListLinePlot[bmPath, AxesLabel -> {"Time", "Location"}, 
  DataRange -> {0, 1}], 
 PlotRange -> 
  Dynamic[{Clip[{.5 - 2^(-zoom), .5 + 2^(-zoom)}, {0, 1}], {-2, 
     2}}], ImageSize -> {200, 100}]}]],
{{zoom, 1, "horizontal zoom"}, 1, 20, 1, 
Appearance -> {"Labeled", "Open"}},
ControlPlacement -> Right, SynchronousUpdating -> True]

Answer 1

It appears that ListLinePlot needs the PlotRange to be an actual list of numbers or another valid setting; something of the form PlotRange -> Dynamic[expr] generates an error. Here's an example:

Module[{bmPath}, bmPath = {0, 2, 1};
 Show[
  ListLinePlot[bmPath, AxesLabel -> {"Time", "Location"}, 
   DataRange -> {0, 1}, ImageSize -> {200, 100}, 
   PlotRange -> 
    Dynamic@{Clip[{.5 - 2^(-2), .5 + 2^(-2)}, {0, 1}], {-2, 2}}]
  ]]

ListLinePlot::prng: Value of option PlotRange -> {{0.25,0.75},{-2,2}} is not All, Full, Automatic, a positive machine number, or an appropriate list of range specifications. >>

ListLinePlot seems to use PlotRange to compute things like the axes, and must have an actual numerical setting or one like Automatic. The way around is to take the graphics returned by ListLinePlot and reset the PlotRange with Show. Graphics may have a dynamic PlotRange. ListLinePlot also explicitly sets the AxesOrigin, so that needs to be reset to Automatic with Show as well.

Manipulate[
 Module[{bmPath},
  bmPath = Accumulate@RandomReal[{-0.1, 0.1}, 10000];
  Row[{
    Show[
     ListLinePlot[bmPath, AxesLabel -> {"Time", "Location"}, 
      DataRange -> {0, 1}, ImageSize -> {200, 100}],
     PlotRange -> Dynamic@{Clip[{.5 - 2^(-zoom), .5 + 2^(-zoom)}, {0, 1}], Automatic},
     PlotRangePadding -> 0,
     AxesOrigin -> Automatic
     ]}]
  ],
 {{zoom, 1, "horizontal zoom"}, 1, 20, 1, Appearance -> {"Labeled", "Open"}},
 ControlPlacement -> Right, SynchronousUpdating -> True]

Answer 2

Let me quote something:

Kuba said:

Why some Options works with Dynamic and other doesn't?

John Fultz said:

The difference is that some option values get translated directly into FE option values. Others affect kernel computation, but don't have a direct translation into the FE.

And, as I've lectured many times before, Dynamic is useless unless it's displaying, directly, in the FE.

For example, look at the Plot function. Something like ImageSize goes directly to the FE. So Dynamic translates well there. But PlotPoints affects the underlying mechanisms of Plot itself, not anything the FE is actually doing.

If you want to change PlotPoints based upon a Dynamic setting, then you need to recompute the entire Plot. So Dynamic needs to wrap the entire Plot.

*FE -> FrontEnd

Answer 3

The problem is that the PlotRangePadding of your ListLinePLot is absolute and doesn't use Scaled expressions.

Compare the behavior of these two examples:

center = 2;
width = 2;
llp = ListLinePlot[{1, 2, 3}];
p = Plot[x, {x, 1, 3}];

Manipulate[
 Show[
  llp,
  PlotRange -> Dynamic@{center + width {-1, 1} 2^-zoom, {1, 3}}
  ],
 {zoom, 1, 20, 1}
 ]

Manipulate[
 Show[
  p,
  PlotRange -> Dynamic@{center + width {-1, 1} 2^-zoom, {1, 3}}
  ],
 {zoom, 1, 20, 1}
 ]

Like your example, the first fails to zoom any further after a certain point. The second works fine.

Take a look at the option settings for each of those plots:

In[90]:= Options[llp, PlotRangePadding]

Out[90]= {PlotRangePadding -> {{0.04, 0.04}, {0.04, 0.04}}}

In[91]:= Options[p, PlotRangePadding]

Out[91]= {PlotRangePadding -> {Scaled[0.02], Scaled[0.02]}}

ref/PlotRangePadding:

PlotRangePadding is an option for graphics functions that specifies how much further axes etc. should extend beyond the range of coordinates specified by PlotRange.

So the behavior here is that your PlotRangePadding setting is keeping your range at a minimum of just over .08 coordinate units despite the PlotRange setting. You'll want to scale that padding as you zoom, thus Scaled is called for.

You might try setting PlotRangePadding in the ListLinePlot expression itself. For some reason, that doesn't seem to work. In 9.0.1 on Linux, I get back the exact same absolute setting.

It seems like you have to set it with Show, side-by-side with PlotRange. E.g.:

Manipulate[
 Show[
  llp,
  PlotRangePadding -> {[email protected], [email protected]},
  PlotRange -> Dynamic@{center + width {-1, 1} 2^-zoom, {1, 3}}
  ],
 {zoom, 1, 20, 1}
 ]

I made the same adjustment to your original example. It worked.