data visualization on a lattice grid

Question

Is it possible to visualize an array whose elements are angles as actual angles on a lattice grid? E.g., say, if the input is:

{{0, π, π}, {0, 0, π/2}, {π/2, 0, 3 π/3}}

then the graphical visualization of this array would be a 3x3 square lattice grid. On each node of the lattice, there will be an arrow (say, inside a circle) which shows the corresponding value from the array. In the above input, this would mean that the arrow at the first lattice-node would be 0, the second would be π, the third would be π, the fourth (i.e., the first node in the second raw of the lattice grid) would be 0, fifth 0, sixth π/2, etc.

Answer 1

It seems natural to convert each number into an arrow (representing a bearing, not an angle) by mapping an arrow-making function over the array. Placing the result into a GraphicsGrid provides options to visualize the lattice if desired.

angles = {{0, π, π}, {0, 0, π/2}, {π/2, 0, 3 π/3}};
GraphicsGrid[
 Map[Graphics[{
    LightGray, Circle[{0, 0}, 1], 
    Hue[#/(2 π), .6, .8], Thick, Arrowheads[Medium], Arrow[{{0, 0}, {Cos[#], Sin[#]}}]}] &, 
  angles, {2}]

Make it fancier, if you like, by souping up the arguments to Graphics. Here I have added another graphical element, hue, to assist in the visual discrimination.

---

As other answers accumulate it is apparent some explanation of this one may be helpful. When the matrix is small, there's not much objectionable to using complex, highly-decorated, or garish (saturated, heavily painted) graphics: chacun a son gout. For larger matrices, though, the point to replacing values with "glyphs," such as arrows, is to be able to visualize patterns. Cleveland et al., Tufte, MacEachren, and many others have shown through theory and experiment how to create effective visualizations: that is, ones that are accurately, quickly, and quantitatively interpreted. Their principles more or less amount to removing inessential material (Tufte's data-ink ratio maximization principle is explicit about this) and representing the numbers using effective graphic symbolization. The method offered in this answer is intended to show how to create effective visualizations on larger data quickly. To see the possibilities, I offer this variation of the problem applied to a larger matrix:

angles = 2 π ImageData[ImageAdjust[Blur[RandomImage[1, {40, 25}], 8], 4]];
GraphicsGrid[
 Map[Graphics[{Hue[#/(2 π), .6, .8], Thin, Arrowheads[Small], 
   Arrow[{{0, 0}, {Cos[#], Sin[#]}}]}] & , angles, {2}]];
Rasterize[%, ImageSize -> 600]

Answer 2

There is a new function AngularGauge in Mathematica 9, which might fit your requirement exactly.

chartFunc[num_] := AngularGauge[num, {0, 2 π},
  ScaleOrigin -> {0, 2 π},
  LabelStyle -> None,
  TicksStyle -> GrayLevel[.3],
  ScaleDivisions -> {36, 5},
  GaugeMarkers -> "ShinyHubNeedle",
  GaugeFrameElementFunction -> ({} &),
  ScaleRanges -> Partition[
    Append[Riffle[
      Partition[Range[0, 2 π, π/6], 2, 1],
      {{-.05, 0}, {-.1, 0}}], {-0.1, 0}],
    2, 2],
  ScaleRangeColorFunction -> (Lighter[ColorData["Rainbow"][#]] &)
  ]

Map[chartFunc, {{0, π, π}, {0, 0, π/2}, {π/2, 0, 3 π/3}}, {2}] // Grid

You can change the GaugeMarkers option to get many other kinds of needles.

Edit

As the code above is how I understand OP's question, that OP might actually want some fancy indicators which are not quantitative but qualitative, inspired by whuber's answer, and noticed that OP is using Mathematica 7.2, here is another built-in function ListVectorPlot, which is first introduced in version 7, for the required purpose:

chartFunc2[data_, size_] := ListVectorPlot[
  Map[{Cos[#], Sin[#]} &, Reverse[data]\[Transpose], {2}],
  VectorColorFunction -> Function[{x, y, vx, vy, n},
    Hue[
     If[vy < .5, 1 - #, #] &[ArcCos[2 vx - 1]/(2 π)],
     .6, .8]],
  VectorStyle -> "CircleArrow",
  VectorScale -> {size, 1, None},
  VectorPoints -> All,
  AspectRatio -> Automatic]

Usage:

chartFunc2[{{0, π, π}, {0, 0, π/2}, {π/2, 0, 3 π/3}}, .3]

angles = 2 π ImageData[ImageAdjust[Blur[RandomImage[1, {40, 25}], 8], 4]];
chartFunc2[angles, .015]

Answer 3

First you have to define a function that will create the arrow inside the circle for every given angle:

arr[th_] := Show[PolarPlot[1, {θ, 0, 2 π}, PlotStyle -> Red, Axes -> None], 
                 Graphics[{Blue, Thick, Arrow[{{0, 0}, Normalize[{Cos[th], Sin[th]}]}]}]]

then, using Map you apply that function to each coefficient of the matrix :

Map[arr, {{0, π, π}, {0, 0,π/2}, {π/2, 0, 3π/3}}, {2}]//TableForm

---

Following David Carraher's comment, if you want angles you can use the function

arr2[th_] := PieChart[{Mod[th, 2 π/(2 π), (2 π - Mod[th, 2 π])/(2 π)},
                      ChartStyle -> {Cyan, LightGray}, SectorOrigin -> 0]

Now,

Map[arr2, {{0, π, π}, {0, 0,π/2}, {π/2, 0, 3π/3}}, {2}]//TableForm

and combining both:

arr3[th_] := Show[PieChart[{Mod[th, 2 π]/(2 π), (2 π - Mod[th, 2 π])/(2 π)}, ChartStyle -> {Cyan, LightGray}, SectorOrigin -> 0,    
                  ChartStyle -> Graphics[{Blue, Thick, Arrow[{{0, 0}, {1, 0}}]}]],   
           Graphics[{{Blue, Thick, Arrow[{{0, 0}, {Cos[th], Sin[th]}}]}, 
           {Blue, Thick, Arrow[{{0, 0}, {1, 0}}]}}]]

Map[arr3, {{0, π, π}, {0, 0,π/2}, {π/2, 0, 3π/3}}, {2}]//TableForm

Answer 4

MapIndexed gives coordinates for drawing on a single canvas.

disks[angles_List] :=
 Graphics[MapIndexed[{
     Orange, Disk[#2, .3],
     Blue, Disk[#2, .3, {0, #1}]} &, angles, {2}]]

disks[{{0, Pi, Pi}, {0, 0, Pi/2}, {Pi/2, 0, 3 Pi/3}}]

disks[RandomReal[{0, 2 Pi}, {3, 3}]]

---

Following @whuber, and using a built-in function:

angles = 2 Pi ImageData[
    ImageAdjust[Blur[RandomImage[1, {40, 25}], 8], 4]];

ListVectorPlot[
 Map[{Cos[#], Sin[#]} &, Transpose@angles, {2}],
 VectorPoints -> All,
 VectorScale -> .02,
 AspectRatio -> Automatic]