Re: ArrayPlot coordinates scaling for overlays

*To*: mathgroup at smc.vnet.net*Subject*: [mg109267] Re: ArrayPlot coordinates scaling for overlays*From*: "David Park" <djmpark at comcast.net>*Date*: Tue, 20 Apr 2010 05:49:04 -0400 (EDT)

With plot functions, WRI usually only lists the options that are special to that plot on the Function help page. You can get the full list of options by using: Options[ArrayPlot] You asked if there was a neater way to do this. My apologies in advance if this does not really interest you but Presentations has rather extensive capabilities for the graphical representation of complex functions and I assume that is your actual objective here. (And you might also get suggestions for making nice complex plots within regular Mathematica.) This material should also appear within a few days in an archive kindly maintained by Peter Lindsay of the Mathematics Department at St. Andrew`s University. It contains the Mathematica notebooks addressing some problem and a PDF for those who don't have Presentations or Mathematica. http://blackbook.mcs.st-and.ac.uk/~Peter/djmpark/html/ Needs["Presentations`Master`"] First, here is your plot done in Presentations. Instead of making separate plots and combining them with a Show statement. You can simply combine all the graphical elements in a single Draw2D statement. arraydata = Table[With[{z = x + I y}, With[{w = (1 + z + z^2 + z^3)/(1 + z + z^2)}, Abs[w]]], {y, -1, 1, 0.1}, {x, -1, 1, 0.1}]; Draw2D[ {Opacity[.5], ArrayDraw[arraydata, ColorFunction -> "TemperatureMap", DataRange -> {{-1, 1}, {-1, 1}}], Opacity[1], Arrowheads[Medium], Arrow[{{0, 0}, {1, 1}}]}, ImageSize -> 250] Presentations has a feature, OptionsFinder, making it easy to get option information and insert it into functions. This was contributed by Thomas Munch and Syd Geraghty. In the above statement, you can place your cursor after ArrayDraw and click the OptionsFinder. This brings up a new palette that contains all the options for ArrayDraw. It has tooltips that show the default value for each option. It also has a link to the help page for the option and a tooltip giving the option usage message. It also has a link to the function. If you click an option, it will be pasted into the statement at the end and the default option value will be highlighted so one can immediately type the desired option value. This is all very useful for using options and getting into help for a function and its options. It is also a little simpler in Presentations because ArrayDraw has fewer options than ArrayPlot, containing only those options that affect the resulting primitives and not the overall plot. Next let's look at a couple of graphical representations of the function. The first representation is a contour plot of the modulus of the function. The ComplexCartesianContour primitive was used to draw the function. Specific contour values are chosen so that the modulus is well represented around both the poles and the zeros. Since the contour spacing's are distinctly non-uniform, the Presentations ContourColors color function is used to give distinct colors for each contour region. (Otherwise most of the graphic would just be blue with a small red region as above.) Notice that we need only one iterator with the min and max values specified by complex numbers. I adjusted the domain to display the region of most interest. The contours have tooltips displaying the contour values. With[ {f = Function[z, (1 + z + z^2 + z^3)/(1 + z + z^2)], zmin = -1.5 (1.25 + I), zmax = (.8 + 1.5 I), contourlist = {0, 0.1, .3, .5, .7, .9, 1, 1.4, 2, 3, 5, 10, 100}}, Draw2D[ {ComplexCartesianContour[f[z], {z, zmin, zmax}, Abs, Contours -> contourlist, ColorFunctionScaling -> False, ColorFunction -> (ContourColors[contourlist, ColorData["TemperatureMap"]]), MaxRecursion -> 3, PlotRange -> {0, 100}]}, Frame -> True, FrameLabel -> {Re, Im}, RotateLabel -> False, PlotLabel -> Row[{"Modulus of ", f[z]}], ImageSize -> 250] ] In the second graphic we use only the contour lines from the previous graphic and place these on top of a domain coloring graphic that shows the argument of the function. The argument varies from -\[Pi] (yellow) to \[Pi] (brown). As an extra detail, the poles are marked with red points and the zeros with blue points. Again the contour lines have tool tips because they actually are from a contour plot. poles = z /. Solve[1 + z + z^2 == 0]; zeros = z /. Solve[1 + z + z^2 + z^3 == 0]; With[ {f = Function[z, (1 + z + z^2 + z^3)/(1 + z + z^2)], zmin = -1.5 (1.25 + I), zmax = (.8 + 1.5 I), contourlist = {0, 0.1, .3, .5, .7, .9, 1, 1.4, 2, 3, 5, 10, 100}, colorfunction = ArgColor[Yellow, Brown, Black, White][]}, Draw2D[ {DomainColoring[f[z], {z, zmin, zmax}, colorfunction, PlotPoints -> 100], ComplexCartesianContour[f[z], {z, zmin, zmax}, Abs, Contours -> contourlist, ContourStyle -> GrayLevel[.3], ContourShading -> False, MaxRecursion -> 3, PlotRange -> {0, 100}], ComplexCirclePoint[#, 3, Black, Red] & /@ poles, ComplexCirclePoint[#, 3, Black, Blue] & /@ zeros}, Frame -> True, FrameLabel -> {Re, Im}, RotateLabel -> False, PlotLabel -> Row[{f[z]}], ImageSize -> 250] ] David Park djmpark at comcast.net http://home.comcast.net/~djmpark/ From: fd [mailto:fdimer at gmail.com] All I've used the DataRange option and it worked quite well. A bit of a downside DataRange is not listed in the options of ArrayPlot. Below is the code with an example of what I'm trying to do. Please let me know if any of you would have a neater way of doing it. Many thanks for the help. a = ArrayPlot[ Table[With[{z = x + I y}, With[{w = (1 + z + z^2 + z^3)/(1 + z + z^2)}, Abs[w]]], {y, -1, 1, 0.1}, {x, -1, 1, 0.1}], ColorFunction -> "TemperatureMap", DataRange -> {{-1, 1}, {-1, 1}}] b = Graphics[Arrow[{{0, 0}, {1, 1}}]] Show[b, Graphics[{Opacity[0.5], First@a}]] On Apr 16, 7:53 pm, Patrick Scheibe <psche... at trm.uni-leipzig.de> wrote: > Hi, > > use DataRange to tell Mathematica about the rectangle where your > ArrayPlot is in > > img = Import["http://sipi.usc.edu/database/misc/5.1.12.tiff";]; > Show[{ > ArrayPlot[img[[1]], DataRange -> {{0, 2 Pi}, {-1, 1}}, > ColorFunction -> GrayLevel], > Plot[Sin[x], {x, 0, 2 Pi}] > }] > > and you can overlay different plots. > > Cheers > Patrick > > > > On Wed, 2010-04-14 at 23:13 -0400, fd wrote: > > All > > > This seems a simple problem I not finding an easy solution. > > > I have a plot obtained from an ArrayPlot, for which the coordinates > > are the indexes of the matrix being plotted; I want to overlay to this > > plot some other plot, say, from DensityPlot. I have to tell > > Mathematica that the bottom left corner of the ArrayPlot is {xi,yi} > > and the upper right is {xf,yf}. > > > It would be nice as well to know how you could do this with a raster > > image in general. > > > I was trying to use ListDensityPlot, but for the specific problem I > > dealing with it is excruciatingly slow. > > > I'm also working to re-scale the FrameTicks by defining a new > > ArrayPlot function, with limited success. Below the code I'm working > > on. > > > Would anyone have an idea about this? Thanks in advance for any help. > > Felipe > > > arrayPlotScale[array_List, {xmin_, xmax_}, {ymin_, ymax_}] := > > Module[{deltas = > > Reverse[{ymax - ymin, xmax - xmin}/Dimensions[array]], > > n = Dimensions[array] // Reverse}, > > ArrayPlot[array, > > FrameTicks -> > > Reverse[{Table[{i, xmin + i deltas[[1]] }, {i, 0, n[[1]], 20}], > > Table[{n[[2]] - i, ymin + i deltas[[2]]}, {i, 0, n[[2]], > > 10}]}]]] > > > test = Table[i j, {i, 1, 100}, {j, 100, 1, -1}]; > > > arrayPlotScale[test, {0, 16}, {0, 100}]