--- title: "DensityPlot" language: "en" type: "Symbol" summary: "DensityPlot[f, {x, xmin, xmax}, {y, ymin, ymax}] makes a density plot of f as a function of x and y. DensityPlot[f, {x, y} \\[Element] reg] takes the variables {x, y} to be in the geometric region reg." keywords: - continuous intensity - continuous density - false color plots - intensity plot - smooth density - scalar field - heat map - densityplot canonical_url: "https://reference.wolfram.com/language/ref/DensityPlot.html" source: "Wolfram Language Documentation" related_guides: - title: "Function Visualization" link: "https://reference.wolfram.com/language/guide/FunctionVisualization.en.md" - title: "Solvers over Regions" link: "https://reference.wolfram.com/language/guide/GeometricSolvers.en.md" related_functions: - title: "ListDensityPlot" link: "https://reference.wolfram.com/language/ref/ListDensityPlot.en.md" - title: "DensityPlot3D" link: "https://reference.wolfram.com/language/ref/DensityPlot3D.en.md" - title: "SliceDensityPlot3D" link: "https://reference.wolfram.com/language/ref/SliceDensityPlot3D.en.md" - title: "ContourPlot" link: "https://reference.wolfram.com/language/ref/ContourPlot.en.md" - title: "ArrayPlot" link: "https://reference.wolfram.com/language/ref/ArrayPlot.en.md" - title: "Plot3D" link: "https://reference.wolfram.com/language/ref/Plot3D.en.md" - title: "StreamPlot" link: "https://reference.wolfram.com/language/ref/StreamPlot.en.md" - title: "VectorPlot" link: "https://reference.wolfram.com/language/ref/VectorPlot.en.md" - title: "StreamDensityPlot" link: "https://reference.wolfram.com/language/ref/StreamDensityPlot.en.md" - title: "VectorDensityPlot" link: "https://reference.wolfram.com/language/ref/VectorDensityPlot.en.md" - title: "ColorFunction" link: "https://reference.wolfram.com/language/ref/ColorFunction.en.md" - title: "RegionPlot" link: "https://reference.wolfram.com/language/ref/RegionPlot.en.md" - title: "ComplexPlot" link: "https://reference.wolfram.com/language/ref/ComplexPlot.en.md" related_tutorials: - title: "Density and Contour Plots" link: "https://reference.wolfram.com/language/tutorial/TheStructureOfGraphicsAndSound.en.md#14270" --- # DensityPlot DensityPlot[f, {x, xmin, xmax}, {y, ymin, ymax}] makes a density plot of f as a function of x and y. DensityPlot[f, {x, y}∈reg] takes the variables {x, y} to be in the geometric region reg. ## Details and Options * ``DensityPlot`` is also known as heat map. * It evaluates ``f`` at values of ``x`` and ``y`` in the domain being plotted over and uses a color function $c$ to map each value ``f[x, y]`` to a color. * The plot visualizes the set $\{\{x,y,c(f(x,y))\},\{x,y\}\in \text{reg}\}$ where $c$ is a color function mapping $f$-values to colors. [image] * At positions where ``f`` does not evaluate to a real number, holes are left so that the background to the density plot shows through. * ``DensityPlot`` treats the variables ``x`` and ``y`` as local, effectively using ``Block``. * ``DensityPlot`` has attribute ``HoldAll``, and evaluates ``f`` only after assigning specific numerical values to ``x`` and ``y``. * In some cases, it may be more efficient to use ``Evaluate`` to evaluate ``f`` symbolically before specific numerical values are assigned to ``x`` and ``y``. * ``DensityPlot`` has the same options as ``Graphics``, with the following additions and changes: [] | | | | | :-------------------- | :---------------------- | :---------------------------------------------------------------------- | | AspectRatio | 1 | ratio of height to width | | BoundaryStyle | None | how to draw RegionFunction boundaries | | BoxRatios | Automatic | effective 3D bounding box ratios | | ClippingStyle | None | how to draw values clipped by PlotRange | | ColorFunction | Automatic | how to color the plot | | ColorFunctionScaling | True | whether to scale the argument to ColorFunction | | EvaluationMonitor | None | expression to evaluate at every function evaluation | | Exclusions | Automatic | x, y curves to exclude | | ExclusionsStyle | None | what to draw at excluded curves | | Frame | True | whether to draw a frame around the plot | | FrameTicks | Automatic | frame tick marks | | LightingAngle | None | effective angle of the simulated light source | | MaxRecursion | Automatic | the maximum number of recursive subdivisions allowed | | Mesh | None | how many mesh lines in each direction to draw | | MeshFunctions | {#1&, #2&} | how to determine the placement of mesh lines | | MeshStyle | Automatic | the style for mesh lines | | Method | Automatic | the method to use for refining the plot | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotLayout | Automatic | how to position densities | | PlotLegends | None | legends for color gradients | | PlotPoints | Automatic | the initial number of sample points for the function in each direction | | PlotRange | {Full, Full, Automatic} | the range of f or other values to include | | PlotRangeClipping | True | whether to clip at the plot range | | PlotRangePadding | Automatic | how much to pad the range of values | | PlotTheme | \$PlotTheme | overall theme for the plot | | RegionFunction | (True&) | how to determine whether a point should be included | | ScalingFunctions | None | how to scale individual coordinates | | WorkingPrecision | MachinePrecision | the precision used in internal computations | * Typical settings for ``PlotLegends`` include: | | | | ---------------- | ------------------------------------------------- | | None | no legend | | Automatic | automatically determine legend from ColorFunction | | Placed[lspec, …] | specify placement for legend | * Possible settings for ``PlotLayout`` that show single densities in multiple plot panels include: | | | | ------------------------------------- | -------------------------------------------- | | "Column" | use separate densities in a column of panels | | "Row" | use separate densities in a row of panels | | {"Column", k}, {"Row", k} | use k columns or rows | | {"Column", UpTo[k]}, {"Row", UpTo[k]} | use at most k columns or rows | [image] * ``DensityPlot`` initially evaluates ``f`` at a grid of equally spaced sample points specified by ``PlotPoints``. Then it uses an adaptive algorithm to subdivide at most ``MaxRecursion`` times to generate smooth contours. * You should realize that since it uses only a finite number of sample points, it is possible for ``DensityPlot`` to miss features of your functions. To check your results, you should try increasing the settings for ``PlotPoints`` and ``MaxRecursion``. * With the setting ``Mesh -> All``, ``DensityPlot`` draws mesh lines to show all the subdivisions it uses. * The default setting ``MeshFunctions -> {#1&, #2&}`` draws an ``x``, ``y`` mesh. * The arguments supplied to functions in ``MeshFunctions`` and ``RegionFunction`` are ``x``, ``y``, ``f``. * ``ColorFunction`` is supplied with a single argument, given by default by the scaled value of ``f``. * With the default settings ``Exclusions -> Automatic`` and ``ExclusionsStyle -> None``, ``DensityPlot`` breaks continuity in the density it displays at any discontinuity curve it detects. * Possible settings for ``ScalingFunctions`` include: | | | | ------------ | ------------------------------- | | sf | scale the f values | | {sx, sy} | scale x and y axes | | {sx, sy, sf} | scale x and y axes and f values | * Each scaling function ``si`` is either a string ``"scale"`` or ``{g, g^-1}`` where ``g^-1`` is the inverse of ``g``. * ``DensityPlot`` returns ``Graphics[GraphicsComplex[data]]``. ### List of all options | | | | | ---------------------- | ----------------------- | ---------------------------------------------------------------------------------- | | AlignmentPoint | Center | the default point in the graphic to align with | | AspectRatio | 1 | ratio of height to width | | Axes | False | whether to draw axes | | AxesLabel | None | axes labels | | AxesOrigin | Automatic | where axes should cross | | AxesStyle | {} | style specifications for the axes | | Background | None | background color for the plot | | BaselinePosition | Automatic | how to align with a surrounding text baseline | | BaseStyle | {} | base style specifications for the graphic | | BoundaryStyle | None | how to draw RegionFunction boundaries | | BoxRatios | Automatic | effective 3D bounding box ratios | | ClippingStyle | None | how to draw values clipped by PlotRange | | ColorFunction | Automatic | how to color the plot | | ColorFunctionScaling | True | whether to scale the argument to ColorFunction | | ContentSelectable | Automatic | whether to allow contents to be selected | | CoordinatesToolOptions | Automatic | detailed behavior of the coordinates tool | | Epilog | {} | primitives rendered after the main plot | | EvaluationMonitor | None | expression to evaluate at every function evaluation | | Exclusions | Automatic | x, y curves to exclude | | ExclusionsStyle | None | what to draw at excluded curves | | FormatType | TraditionalForm | the default format type for text | | Frame | True | whether to draw a frame around the plot | | FrameLabel | None | frame labels | | FrameStyle | {} | style specifications for the frame | | FrameTicks | Automatic | frame tick marks | | FrameTicksStyle | {} | style specifications for frame ticks | | GridLines | None | grid lines to draw | | GridLinesStyle | {} | style specifications for grid lines | | ImageMargins | 0. | the margins to leave around the graphic | | ImagePadding | All | what extra padding to allow for labels etc. | | ImageSize | Automatic | the absolute size at which to render the graphic | | LabelStyle | {} | style specifications for labels | | LightingAngle | None | effective angle of the simulated light source | | MaxRecursion | Automatic | the maximum number of recursive subdivisions allowed | | Mesh | None | how many mesh lines in each direction to draw | | MeshFunctions | {#1&, #2&} | how to determine the placement of mesh lines | | MeshStyle | Automatic | the style for mesh lines | | Method | Automatic | the method to use for refining the plot | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotLabel | None | an overall label for the plot | | PlotLayout | Automatic | how to position densities | | PlotLegends | None | legends for color gradients | | PlotPoints | Automatic | the initial number of sample points for the function in each direction | | PlotRange | {Full, Full, Automatic} | the range of f or other values to include | | PlotRangeClipping | True | whether to clip at the plot range | | PlotRangePadding | Automatic | how much to pad the range of values | | PlotRegion | Automatic | the final display region to be filled | | PlotTheme | \$PlotTheme | overall theme for the plot | | PreserveImageOptions | Automatic | whether to preserve image options when displaying new versions of the same graphic | | Prolog | {} | primitives rendered before the main plot | | RegionFunction | (True&) | how to determine whether a point should be included | | RotateLabel | True | whether to rotate y labels on the frame | | ScalingFunctions | None | how to scale individual coordinates | | Ticks | Automatic | axes ticks | | TicksStyle | {} | style specifications for axes ticks | | WorkingPrecision | MachinePrecision | the precision used in internal computations | --- ## Examples (132) ### Basic Examples (4) Plot a function: ```wl In[1]:= DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}] Out[1]= [image] ``` --- Use a different color scheme and legend: ```wl In[1]:= DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, ColorFunction -> "SunsetColors", PlotLegends -> Automatic] Out[1]= [image] ``` --- Create a contouring overlay mesh: ```wl In[1]:= DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, ColorFunction -> "CMYKColors", PlotPoints -> 35, MeshFunctions -> {#3&, #3&}, Mesh -> {Range[-1, 1, 0.4], Range[-0.8, 0.8, 0.4]}, MeshStyle -> Opacity[0.5, Black]] Out[1]= [image] ``` --- Use a multi-panel layout to show multiple functions at the same time: ```wl In[1]:= DensityPlot[{Sin[x]Cos[y], Sin[x + y]Cos[x - y]}, {x, -5, 5}, {y, -5, 5}, PlotLayout -> "Row"] Out[1]= [image] ``` ### Scope (19) #### Sampling (11) More points are sampled where the function changes quickly: ```wl In[1]:= DensityPlot[Sin[x y], {x, 0, 3}, {y, 0, 3}, Mesh -> All] Out[1]= [image] ``` --- The plot range is selected automatically: ```wl In[1]:= DensityPlot[1 / (x ^ 2 + y ^ 2), {x, -1, 1}, {y, -1, 1}] Out[1]= [image] ``` --- Areas where the function becomes nonreal are excluded: ```wl In[1]:= DensityPlot[Sqrt[x y], {x, -1, 1}, {y, -1, 1}] Out[1]= [image] ``` --- The region is split when there are discontinuities in the function: ```wl In[1]:= DensityPlot[Im[Sqrt[(x + I y) ^ 3]], {x, -2, 2}, {y, -2, 2}] Out[1]= [image] ``` --- Use ``PlotPoints`` and ``MaxRecursion`` to control adaptive sampling: ```wl In[1]:= Grid[Table[DensityPlot[Sin[x y], {x, 0, 4}, {y, 0, 4}, PlotPoints -> pp, MaxRecursion -> mr, Mesh -> None], {mr, {0, 2}}, {pp, {5, 15}}]] Out[1]= | | | | ------- | ------- | | [image] | [image] | | [image] | [image] | ``` --- Use ``PlotRange`` to focus in on areas of interest: ```wl In[1]:= {DensityPlot[x ^ 4 - 2x ^ 2 + y ^ 4 - 2y ^ 2 + 5, {x, -2, 2}, {y, -2, 2}], DensityPlot[x ^ 4 - 2x ^ 2 + y ^ 4 - 2y ^ 2 + 1, {x, -2, 2}, {y, -2, 2}, PlotRange -> {-2, 2}, ClippingStyle -> None]} Out[1]= {[image], [image]} ``` --- Use ``Exclusions`` to remove curves or split the resulting surface: ```wl In[1]:= f = 1 / ChebyshevU[x y - 1, 2];{DensityPlot[f, {x, -5, 5}, {y, -5, 5}, ClippingStyle -> Automatic], DensityPlot[f, {x, -5, 5}, {y, -5, 5}, ClippingStyle -> Automatic, Exclusions -> {x == 0, y == 0}]} Out[1]= {[image], [image]} ``` --- Use ``RegionFunction`` to restrict the surface to a region given by inequalities: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, RegionFunction -> Function[{x, y, z}, 0 < x + y ^ 2 < Pi]] Out[1]= [image] ``` --- The domain may be specified by a region: ```wl In[1]:= DensityPlot[Sin[x + Cos[y]], {x, y}∈Disk[{0, 0}, 3]] Out[1]= [image] ``` --- The domain may be specified by a ``MeshRegion`` : ```wl In[1]:= \[ScriptCapitalD] = MeshRegion[{{0, 0}, {5, -2}, {3, 0}, {5, 2}}, Polygon[{1, 2, 3, 4}]]; In[2]:= DensityPlot[Sin[x]Cos[y], {x, y}∈\[ScriptCapitalD]] Out[2]= [image] ``` --- Plot over an infinite domain: ```wl In[1]:= DensityPlot[(x y^2/x^2 + y^4), {x, -Infinity, Infinity}, {y, -Infinity, Infinity}] Out[1]= [image] ``` #### Presentation (8) Add labels: ```wl In[1]:= DensityPlot[Sin[x y], {x, 0, 3}, {y, 0, 3}, FrameLabel -> {x, y}, PlotLabel -> Sin[x y], ColorFunction -> "SolarColors"] Out[1]= [image] ``` --- Color the surface by height: ```wl In[1]:= DensityPlot[Sin[x y], {x, 0, 3}, {y, 0, 3}, ColorFunction -> GrayLevel] Out[1]= [image] In[2]:= DensityPlot[Sin[x y], {x, 0, 3}, {y, 0, 3}, ColorFunction -> "SouthwestColors", PlotPoints -> 35] Out[2]= [image] ``` --- Add a legend: ```wl In[1]:= DensityPlot[Sin[x y], {x, 0, 3}, {y, 0, 3}, ColorFunction -> "SouthwestColors", PlotPoints -> 35, PlotLegends -> Automatic] Out[1]= [image] ``` --- Provide an interactive ``Tooltip`` for a surface: ```wl In[1]:= DensityPlot[Tooltip[Sqrt[x y]], {x, -2, 2}, {y, -2, 2}, ColorFunction -> "DeepSeaColors"] Out[1]= [image] ``` --- Style the overlay mesh: ```wl In[1]:= DensityPlot[Sin[x y], {x, 0, 3}, {y, 0, 3}, Mesh -> 5, ColorFunction -> "DarkRainbow", MeshStyle -> {Purple, Yellow}, PlotPoints -> 35] Out[1]= [image] ``` --- Create a contouring overlay mesh: ```wl In[1]:= DensityPlot[Sin[x y], {x, 0, 3}, {y, 0, 3}, ColorFunction -> "TemperatureMap", PlotPoints -> 35, MeshFunctions -> {#3&, #3&}, Mesh -> {Range[-1, 1, 0.4], Range[-0.8, 0.8, 0.4]}, MeshStyle -> {Black, Dashed}] Out[1]= [image] ``` --- Use a theme with simple ticks in a high-contrast color scheme: ```wl In[1]:= DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, PlotTheme -> "Marketing"] Out[1]= [image] ``` --- Show multiple functions as densities in separate panels: ```wl In[1]:= DensityPlot[{Sin[x]Cos[y], Sin[x + y]Cos[x - y]}, {x, -7, 7}, {y, -7, 7}, PlotLayout -> "Row"] Out[1]= [image] ``` Use a column instead of a row: ```wl In[2]:= DensityPlot[{Sin[x]Cos[y], Sin[x + y]Cos[x - y]}, {x, -7, 7}, {y, -7, 7}, PlotLayout -> "Column"] Out[2]= [image] ``` ### Options (89) #### AspectRatio (4) By default, ``DensityPlot`` uses the same width and height: ```wl In[1]:= DensityPlot[Sin[x]Cos[y], {x, -2Pi, 2Pi}, {y, -Pi / 2, 3Pi / 2}] Out[1]= [image] ``` --- Use a numerical value to specify the height to width ratio: ```wl In[1]:= DensityPlot[Sin[x]Cos[y], {x, -2Pi, 2Pi}, {y, -Pi / 2, 3Pi / 2}, AspectRatio -> 1 / 2] Out[1]= [image] ``` --- ``AspectRatio -> Automatic`` determines the ratio from the plot ranges: ```wl In[1]:= DensityPlot[Sin[x]Cos[y], {x, -2Pi, 2Pi}, {y, -Pi / 2, 3Pi / 2}, AspectRatio -> Automatic] Out[1]= [image] ``` --- ``AspectRatio -> Full`` adjusts the height and width to tightly fit inside other constructs: ```wl In[1]:= plot = DensityPlot[Sin[x]Cos[y], {x, -2Pi, 2Pi}, {y, -Pi / 2, 3Pi / 2}, AspectRatio -> Full]; In[2]:= {Framed[Pane[plot, {50, 100}]], Framed[Pane[plot, {100, 100}]], Framed[Pane[plot, {100, 50}]]} Out[2]= {[image], [image], [image]} ``` #### Axes (4) By default, ``DensityPlot`` uses a frame instead of axes: ```wl In[1]:= DensityPlot[x Sin[y], {x, -5, 5}, {y, -5, 5}] Out[1]= [image] ``` --- Use axes instead of a frame: ```wl In[1]:= DensityPlot[x Sin[y], {x, -5, 5}, {y, -5, 5}, Frame -> False, Axes -> True] Out[1]= [image] ``` --- Use ``AxesOrigin`` to specify where the axes intersect: ```wl In[1]:= DensityPlot[x Sin[y], {x, -5, 5}, {y, -5, 5}, Frame -> False, Axes -> True, AxesOrigin -> {1, 1}] Out[1]= [image] ``` --- Turn each axis on individually: ```wl In[1]:= {DensityPlot[x Sin[y], {x, -5, 5}, {y, -5, 5}, Frame -> False, Axes -> {True, False}], DensityPlot[x Sin[y], {x, -5, 5}, {y, -5, 5}, Frame -> False, Axes -> {False, True}]} Out[1]= {[image], [image]} ``` #### AxesLabel (4) No axes labels are drawn by default: ```wl In[1]:= DensityPlot[Sin[x ] + Cos[y], {x, -5, 5}, {y, -5, 5}, Frame -> False, Axes -> True] Out[1]= [image] ``` --- Place a label on the $y$ axis: ```wl In[1]:= DensityPlot[Sin[x ] + Cos[y], {x, -5, 5}, {y, -5, 5}, Frame -> False, Axes -> True, AxesLabel -> y] Out[1]= [image] ``` --- Specify axes labels: ```wl In[1]:= DensityPlot[Sin[x ] + Cos[y], {x, -5, 5}, {y, -5, 5}, Frame -> False, Axes -> True, AxesLabel -> {"x-label", "y-label"}] Out[1]= [image] ``` --- Use labels based on variables specified in ``DensityPlot`` : ```wl In[1]:= DensityPlot[Sin[x ] + Cos[y], {x, -5, 5}, {y, -5, 5}, Frame -> False, Axes -> True, AxesLabel -> Automatic] Out[1]= [image] ``` #### AxesOrigin (2) The position of the axes is determined automatically: ```wl In[1]:= DensityPlot[Sin[x ] + Cos[y], {x, -5, 5}, {y, -5, 5}, Frame -> False, Axes -> True] Out[1]= [image] ``` --- Specify an explicit origin for the axes: ```wl In[1]:= DensityPlot[Sin[x ] + Cos[y], {x, -5, 5}, {y, -5, 5}, Frame -> False, Axes -> True, AxesOrigin -> {1, 0}] Out[1]= [image] ``` #### AxesStyle (4) Change the style for the axes: ```wl In[1]:= DensityPlot[x Sin[y], {x, -5, 5}, {y, -5, 5}, Frame -> False, Axes -> True, AxesStyle -> Red] Out[1]= [image] ``` --- Specify the style of each axis: ```wl In[1]:= DensityPlot[x Sin[y], {x, -5, 5}, {y, -5, 5}, Frame -> False, Axes -> True, AxesStyle -> {Red, Blue}] Out[1]= [image] ``` --- Use different styles for the ticks and the axes: ```wl In[1]:= DensityPlot[x Sin[y], {x, -5, 5}, {y, -5, 5}, Frame -> False, Axes -> True, AxesStyle -> Green, TicksStyle -> Black] Out[1]= [image] ``` --- Use different styles for the labels and the axes: ```wl In[1]:= DensityPlot[x Sin[y], {x, -5, 5}, {y, -5, 5}, Frame -> False, Axes -> True, AxesStyle -> Green, LabelStyle -> Black] Out[1]= [image] ``` #### BoundaryStyle (3) Use a red boundary around the edges of the surface: ```wl In[1]:= DensityPlot[Sin[x y], {x, -2, 2}, {y, -2, 2}, BoundaryStyle -> Red] Out[1]= [image] ``` --- ``BoundaryStyle`` applies to regions cut by ``RegionFunction`` : ```wl In[1]:= DensityPlot[Sin[x y], {x, -2, 2}, {y, -2, 2}, BoundaryStyle -> Red, RegionFunction -> Function[{x, y, z}, x ^ 2 + y ^ 2 ≥ 1]] Out[1]= [image] ``` --- ``BoundaryStyle`` does not apply to cuts made by ``Exclusions`` : ```wl In[1]:= DensityPlot[Im[Sqrt[x + I y]], {x, -2, 2}, {y, -2, 2}, BoundaryStyle -> Red] Out[1]= [image] ``` Use ``ExclusionsStyle`` instead: ```wl In[2]:= DensityPlot[Im[Sqrt[x + I y]], {x, -2, 2}, {y, -2, 2}, ExclusionsStyle -> Red] Out[2]= [image] ``` #### ClippingStyle (4) Show clipped regions like the rest of the surface: ```wl In[1]:= DensityPlot[Im[Gamma[x + I y]], {x, -3.5, 3.5}, {y, -3.5, 3.5}, ClippingStyle -> Automatic] Out[1]= [image] ``` --- Leave clipped regions empty: ```wl In[1]:= DensityPlot[Im[Gamma[x + I y]], {x, -3.5, 3.5}, {y, -3.5, 3.5}, ClippingStyle -> None] Out[1]= [image] ``` --- Use pink to fill the clipped regions: ```wl In[1]:= DensityPlot[Im[Gamma[x + I y]], {x, -3.5, 3.5}, {y, -3.5, 3.5}, ClippingStyle -> Pink] Out[1]= [image] ``` --- Use light red where the surface is clipped above and pink below: ```wl In[1]:= DensityPlot[Im[Gamma[x + I y]], {x, -3.5, 3.5}, {y, -3.5, 3.5}, ClippingStyle -> {Pink, LightRed}] Out[1]= [image] ``` #### ColorFunction (5) Color by scaled $z$ coordinate: ```wl In[1]:= DensityPlot[x + Sin[x ^ 2 + y ^ 2], {x, -4, 4}, {y, -4, 4}, ColorFunction -> Hue] Out[1]= [image] ``` --- Specify gray-level intensity by scaled $z$ coordinate: ```wl In[1]:= DensityPlot[y + Sin[x ^ 2 + 3y], {x, -3, 3}, {y, -3, 3}, ColorFunction -> GrayLevel] Out[1]= [image] ``` --- Named color gradients color in the $z$ direction: ```wl In[1]:= DensityPlot[y + Sin[x ^ 2 + 3y], {x, -3, 3}, {y, -3, 3}, ColorFunction -> "SunsetColors"] Out[1]= [image] ``` --- Use brightness to correspond to height or density of a function: ```wl In[1]:= DensityPlot[y + Sin[x ^ 2 + 3y], {x, -3, 3}, {y, -3, 3}, ColorFunction -> (Hue[2 / 5, 2 / 3, #]&)] Out[1]= [image] ``` --- Use the interpolation between two colors to indicate the height or density of a function: ```wl In[1]:= DensityPlot[y + Sin[x ^ 2 + 3y], {x, -3, 3}, {y, -3, 3}, ColorFunction -> (RGBColor[1 - #, #, 1]&)] Out[1]= [image] ``` #### ColorFunctionScaling (1) Get the natural range of values by setting ``ColorFunctionScaling`` to ``False`` : ```wl In[1]:= DensityPlot[550 + 150 Sin[x]Sin[y], {x, 0, 2Pi}, {y, 0, 2Pi}, ColorFunction -> (ColorData["VisibleSpectrum"][#1]&), ColorFunctionScaling -> False, PlotPoints -> 50] Out[1]= [image] ``` #### EvaluationMonitor (2) Show where ``DensityPlot`` samples a function: ```wl In[1]:= ListPlot[Reap[DensityPlot[Sin[x]Sin[y], {x, -2, 2}, {y, -3, 3}, EvaluationMonitor :> Sow[{x, y}]]][[-1, 1]]] Out[1]= [image] ``` --- Count how many times $\sin (x y)$ is evaluated: ```wl In[1]:= Block[{k = 0}, DensityPlot[Sin[x y], {x, 0, 2}, {y, 0, 2}, EvaluationMonitor :> k++];k] Out[1]= 400 ``` #### Exclusions (6) This uses automatic methods to compute exclusions: ```wl In[1]:= DensityPlot[Im[ArcSin[x + I y]], {x, -2, 2}, {y, -2, 2}] Out[1]= [image] ``` --- Indicate that no exclusions should be computed: ```wl In[1]:= DensityPlot[Im[ArcSin[x + I y]], {x, -2, 2}, {y, -2, 2}, Exclusions -> None] Out[1]= [image] ``` --- Give exclusions as an equation: ```wl In[1]:= DensityPlot[ 1 / (y ^ 2 - x ^ 3 + 3x - 3), {x, -3, 3}, {y, -3, 3}, Exclusions -> {y ^ 2 - x ^ 3 + 3x - 3 == 0}, ClippingStyle -> Automatic] Out[1]= [image] ``` --- Give multiple exclusion sets: ```wl In[1]:= DensityPlot[Tan[x y] + 1 / (y ^ 2 - x ^ 3 + 3x - 3), {x, -2, 2}, {y, -2, 2}, Exclusions -> {Cos[x y] == 0, y ^ 2 - x ^ 3 + 3x - 3 == 0}, ClippingStyle -> Automatic] Out[1]= [image] ``` --- Use a condition with the exclusion equation: ```wl In[1]:= DensityPlot[Im[Sqrt[x + I y]], {x, -2, 2}, {y, -2, 2}, Exclusions -> {{y == 0, x ≤ 0}}] Out[1]= [image] ``` --- Use both automatically computed and explicit exclusions: ```wl In[1]:= DensityPlot[Im[(x + I y) ^ (1 / 3)] / (x ^ 2 - y ^ 2), {x, -2, 2}, {y, -2, 2}, Exclusions -> {Automatic, x ^ 2 - y ^ 2 == 0}, Mesh -> False, ClippingStyle -> Automatic]//Quiet Out[1]= [image] ``` #### ExclusionsStyle (1) Use a red boundary to indicate the excluded curves: ```wl In[1]:= DensityPlot[Im[ArcSech[ArcSin[x + I y] ^ 5]], {x, -2, 2}, {y, -2, 2}, ExclusionsStyle -> Red] Out[1]= [image] ``` #### ImageSize (7) Use named sizes such as ``Tiny``, ``Small``, ``Medium`` and ``Large`` : ```wl In[1]:= {DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, ImageSize -> Tiny], DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, ImageSize -> Small]} Out[1]= {[image], [image]} ``` --- Specify the width of the plot: ```wl In[1]:= {DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, ImageSize -> 150], DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, AspectRatio -> 1.5, ImageSize -> 150]} Out[1]= {[image], [image]} ``` Specify the height of the plot: ```wl In[2]:= {DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, ImageSize -> {Automatic, 150}], DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, AspectRatio -> 2, ImageSize -> {Automatic, 150}]} Out[2]= {[image], [image]} ``` --- Allow the width and height to be up to a certain size: ```wl In[1]:= {DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, ImageSize -> UpTo[200]], DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, AspectRatio -> 2, ImageSize -> UpTo[200]]} Out[1]= {[image], [image]} ``` --- Specify the width and height for a graphic, padding with space if necessary: ```wl In[1]:= DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, ImageSize -> {300, 200}, Background -> LightBlue] Out[1]= [image] ``` Setting ``AspectRatio -> Full`` will fill the available space: ```wl In[2]:= DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, AspectRatio -> Full, ImageSize -> {300, 200}, Background -> LightBlue] Out[2]= [image] ``` --- Use maximum sizes for the width and height: ```wl In[1]:= {DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, ImageSize -> {UpTo[150], UpTo[100]}], DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, AspectRatio -> 2, ImageSize -> {UpTo[150], UpTo[100]}]} Out[1]= {[image], [image]} ``` --- Use ``ImageSize -> Full`` to fill the available space in an object: ```wl In[1]:= Framed[Pane[DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, ImageSize -> Full, Background -> LightBlue], {200, 100}]] Out[1]= [image] ``` --- Specify the image size as a fraction of the available space: ```wl In[1]:= Framed[Pane[DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, AspectRatio -> Full, ImageSize -> {Scaled[0.5], Scaled[0.5]}, Background -> LightBlue], {200, 100}]] Out[1]= [image] ``` #### MaxRecursion (1) Refine the function where it changes quickly: ```wl In[1]:= Table[DensityPlot[Sin[x y], {x, 0, 3}, {y, 0, 3}, Mesh -> All, Axes -> False, MaxRecursion -> r], {r, {0, 2}}] Out[1]= {[image], [image]} ``` #### Mesh (6) Use no mesh: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, Mesh -> None] Out[1]= [image] ``` --- Show the initial and final sampling mesh: ```wl In[1]:= {DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, Mesh -> Full], DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, Mesh -> All]} Out[1]= {[image], [image]} ``` --- Use 5 mesh lines in each direction: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, Mesh -> 5] Out[1]= [image] ``` --- Use 3 mesh lines in the $x$ direction and 6 mesh lines in the $y$ direction: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, Mesh -> {3, 6}] Out[1]= [image] ``` --- Use mesh lines at specific values: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, Mesh -> {{-1}, {0}}] Out[1]= [image] ``` --- Use different styles for different mesh lines: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, Mesh -> {{{-1, Red}}, {{0, Blue}}}] Out[1]= [image] ``` #### MeshFunctions (3) Use the $z$ value as the mesh function: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, MeshFunctions -> {#3&}, Mesh -> 5] Out[1]= [image] ``` --- Use mesh lines in the $x$ and $y$ directions: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, MeshFunctions -> {#1&, #2&}, Mesh -> {5, 5}] Out[1]= [image] ``` --- Use mesh lines corresponding to fixed distances from the origin: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, MeshFunctions -> {Norm[{#1, #2, #3}]&}, Mesh -> 5] Out[1]= [image] ``` #### MeshStyle (2) Use red mesh lines: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, MeshStyle -> Red, Mesh -> 3] Out[1]= [image] ``` --- Use red mesh lines in the $x$ direction and dashed mesh lines in the $y$ direction: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, MeshStyle -> {Red, Dashed}, Mesh -> 3] Out[1]= [image] ``` #### PerformanceGoal (2) Generate a higher-quality plot: ```wl In[1]:= Timing[DensityPlot[Sin[x + y ^ 2], {x, -2, 2}, {y, -2, 2}, ColorFunction -> "Rainbow", PerformanceGoal -> "Quality"]] Out[1]= {0.015625, [image]} ``` --- Emphasize performance, possibly at the cost of quality: ```wl In[1]:= Timing[DensityPlot[Sin[x + y ^ 2], {x, -2, 2}, {y, -2, 2}, ColorFunction -> "Rainbow", PerformanceGoal -> "Speed"]] Out[1]= {0., [image]} ``` #### PlotLayout (3) Place each density in a separate panel using shared axes: ```wl In[1]:= DensityPlot[{Cos[x], Cos[x + y], Cos[x + y ^ 2]}, {x, -3, 3}, {y, -2, 2}, ImageSize -> Medium, PlotLayout -> "Column"] Out[1]= [image] ``` Use a row instead of a column: ```wl In[2]:= DensityPlot[{Cos[x], Cos[x + y], Cos[x + y ^ 2]}, {x, -3, 3}, {y, -2, 2}, ImageSize -> Medium, PlotLayout -> "Row"] Out[2]= [image] ``` --- Use multiple columns or rows: ```wl In[1]:= DensityPlot[{Cos[x], Cos[x + y], Cos[x + y ^ 2]}, {x, -3, 3}, {y, -2, 2}, ImageSize -> Medium, PlotLayout -> {"Column", 2}] Out[1]= [image] ``` --- Prefer full columns or rows: ```wl In[1]:= DensityPlot[{Sin[x + y], Sin[2 x + y], Sin[3 x + y], Sin[4 x + y], Sin[5 x + y], Sin[6 x + y]}, {x, -3, 3}, {y, -2, 2}, ImageSize -> Medium, PlotLayout -> {"Column", UpTo[4]}] Out[1]= [image] In[2]:= DensityPlot[{Sin[x + y], Sin[2 x + y], Sin[3 x + y], Sin[4 x + y], Sin[5 x + y], Sin[6 x + y]}, {x, -3, 3}, {y, -2, 2}, ImageSize -> Medium, PlotLayout -> {"Column", 4}] Out[2]= [image] ``` #### PlotLegends (4) Show a legend for the heights: ```wl In[1]:= DensityPlot[Sin[2 x + y]Cos[x - 2y], {x, 0, Pi}, {y, 0, Pi}, PlotLegends -> Automatic] Out[1]= [image] ``` --- ``PlotLegends`` automatically matches the color function: ```wl In[1]:= DensityPlot[Sin[2 x + y]Cos[x - 2y], {x, 0, Pi}, {y, 0, Pi}, ColorFunction -> "GrayYellowTones", PlotLegends -> Automatic] Out[1]= [image] ``` --- Use ``Placed`` to change legend position: ```wl In[1]:= Table[DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, PlotLegends -> Placed[Automatic, pos], PlotLabel -> pos, ImageSize -> 150], {pos, {Before, After, Above, Below}}] Out[1]= [image] ``` --- Use ``BarLegend`` to change legend appearance: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, PlotLegends -> BarLegend[Automatic, LegendMarkerSize -> 180, LegendFunction -> "Frame", LegendMargins -> 5, LegendLabel -> "z-value"]] Out[1]= [image] ``` #### PlotPoints (2) Use more initial points to get a smoother density: ```wl In[1]:= Table[DensityPlot[Sin[x + y ^ 2], {x, -3, 3}, {y, -2, 2}, Mesh -> None, MaxRecursion -> 0, PlotPoints -> pp], {pp, {5, 10, 15, 20}}] Out[1]= {[image], [image], [image], [image]} ``` --- Use 20 initial points in the $x$ direction and 5 in the $y$ direction: ```wl In[1]:= DensityPlot[Sin[2x]y, {x, 0, 2Pi}, {y, -1, 1}, PlotPoints -> {20, 5}, Mesh -> Full] Out[1]= [image] ``` #### PlotRange (4) Automatically compute the $z$ range: ```wl In[1]:= DensityPlot[1 / (x ^ 2 + y ^ 2), {x, -2, 2}, {y, -2, 2}] Out[1]= [image] ``` --- Use all points to compute the range: ```wl In[1]:= DensityPlot[1 / (x ^ 2 + y ^ 2), {x, -2, 2}, {y, -2, 2}, PlotRange -> All] Out[1]= [image] ``` --- Show the surface over the full $x$, $y$ range: ```wl In[1]:= DensityPlot[Sqrt[1 - x ^ 2 - y ^ 2], {x, -2, 2}, {y, -2, 2}, PlotRange -> Full] Out[1]= [image] ``` Automatically compute the $x$, $y$ range: ```wl In[2]:= DensityPlot[Sqrt[1 - x ^ 2 - y ^ 2], {x, -2, 2}, {y, -2, 2}, PlotRange -> Automatic] Out[2]= [image] ``` --- Use an explicit $z$ range to emphasize features: ```wl In[1]:= {DensityPlot[x ^ 2 - y ^ 2, {x, -4, 4}, {y, -4, 4}], DensityPlot[x ^ 2 - y ^ 2, {x, -4, 4}, {y, -4, 4}, PlotRange -> {-2, 2}]} Out[1]= {[image], [image]} ``` #### PlotTheme (1) Use a theme with detailed ticks and a legend: ```wl In[1]:= DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, PlotTheme -> "Detailed"] Out[1]= [image] ``` Change the color function: ```wl In[2]:= DensityPlot[Sin[x]Sin[y], {x, -4, 4}, {y, -3, 3}, PlotTheme -> "Detailed", ColorFunction -> "SolarColors"] Out[2]= [image] ``` #### RegionFunction (3) Plot over an annulus region in $x$ and $y$ : ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -2, 2}, {y, -2, 2}, RegionFunction -> Function[{x, y, z}, 2 < x ^ 2 + y ^ 2 < 5], BoundaryStyle -> Red] Out[1]= [image] ``` --- Regions do not have to be connected: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -2, 2}, {y, -2, 2}, RegionFunction -> Function[{x, y, z}, 0 < Mod[x ^ 2 + y ^ 2, 2] < 1], BoundaryStyle -> Red] Out[1]= [image] ``` --- Use any logical combination of conditions: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, -2, 2}, {y, -2, 2}, RegionFunction -> Function[{x, y, z}, (x - 1 / 2) ^ 2 + y ^ 2 > 1 && (x + 1 / 2) ^ 2 + y ^ 2 > 1], BoundaryStyle -> Red] Out[1]= [image] ``` #### ScalingFunctions (9) By default, plots have linear scales in each direction: ```wl In[1]:= DensityPlot[Max[x, y], {x, 0, 10}, {y, 0, 10}, Exclusions -> None] Out[1]= [image] ``` --- Use a log scale in the $y$ direction: ```wl In[1]:= DensityPlot[Max[x, y], {x, 0, 10}, {y, 0, 10}, ScalingFunctions -> {None, "Log"}, Exclusions -> None] Out[1]= [image] ``` --- Use a linear scale in the $y$ direction that shows smaller numbers at the top: ```wl In[1]:= DensityPlot[Max[x, y], {x, 0, 10}, {y, 0, 10}, ScalingFunctions -> {None, "Reverse"}, Exclusions -> None] Out[1]= [image] ``` --- Use a reciprocal scale in the $y$ direction: ```wl In[1]:= DensityPlot[Max[x, y], {x, 0, 10}, {y, 0, 10}, ScalingFunctions -> {None, "Reciprocal"}, PlotRangePadding -> None, Exclusions -> None] Out[1]= [image] ``` --- Use different scales in the $x$ and $y$ directions: ```wl In[1]:= DensityPlot[Max[x, y], {x, 0, 10}, {y, 0, 10}, ScalingFunctions -> {"Reverse", "Log"}, Exclusions -> None] Out[1]= [image] ``` --- Reverse the $x$ axis without changing the $y$ axis: ```wl In[1]:= DensityPlot[Max[x, y], {x, 0, 10}, {y, 0, 10}, ScalingFunctions -> {"Reverse", None}, Exclusions -> None] Out[1]= [image] ``` --- Use a scale defined by a function and its inverse: ```wl In[1]:= DensityPlot[Max[x, y], {x, 0, 10}, {y, 0, 10}, ScalingFunctions -> {None, {-Log[#]&, Exp[-#]&}}, Exclusions -> None] Out[1]= [image] ``` --- Positions in ``Ticks`` and ``GridLines`` are automatically scaled: ```wl In[1]:= DensityPlot[Sin[x + y ^ 2], {x, 0.1, 2}, {y, -2, 2}, RegionFunction -> Function[{x, y, z}, 2 < x ^ 2 + y ^ 2 < 5], ScalingFunctions -> {"Log", None}, FrameTicks -> {{Automatic, Automatic}, {2 ^ Range[-5, 2], Automatic}}, GridLines -> All] Out[1]= [image] ``` --- ``PlotRange`` is automatically scaled: ```wl In[1]:= DensityPlot[x ^ 2 + y ^ 2, {x, 0, 10}, {y, 0, 10}, ScalingFunctions -> {None, "Log"}, PlotRange -> {1, 100}] Out[1]= [image] ``` #### WorkingPrecision (2) Evaluate functions using machine-precision arithmetic: ```wl In[1]:= DensityPlot[Sin[x y + 10 ^ 16], {x, 0, 3}, {y, 0, 3}, WorkingPrecision -> MachinePrecision] Out[1]= [image] ``` --- Evaluate functions using arbitrary-precision arithmetic: ```wl In[1]:= DensityPlot[Sin[x y + 10 ^ 16], {x, 0, 3}, {y, 0, 3}, WorkingPrecision -> 20] Out[1]= [image] ``` ### Applications (7) Plot a sum of 5 sine waves in random directions: ```wl In[1]:= DensityPlot[Evaluate[Sum[Sin[RandomReal[5, 2].{x, y}], {5}]], {x, 0, 5}, {y, 0, 5}, ColorFunction -> "Rainbow", PlotPoints -> 50] Out[1]= [image] ``` --- This shows the solution to the heat equation in one dimension: ```wl In[1]:= NDSolve[{D[u[t, x], t] == D[u[t, x], x, x], u[0, x] == 0, u[t, 0] == Sin[t], u[t, 5] == 0}, u, {t, 0, 10}, {x, 0, 5}] Out[1]= {{u -> InterpolatingFunction[{{0., 10.}, {0., 5.}}, {5, 5, 1, {78, 25}, {4, 6}, 0, 0, 0, 0, Automatic, {}, {}, False}, {CompressedData["«923»"], {0., 0.20833333333333334, 0.4166666666666667, 0.625, 0.8333333333333334, 1.0416666666666667, 1.2 ... 3333333333335, 3.5416666666666665, 3.75, 3.9583333333333335, 4.166666666666667, 4.375, 4.583333333333333, 4.791666666666667, 5.}}, {Developer`PackedArrayForm, CompressedData["«3619»"], CompressedData["«39443»"]}, {Automatic, Automatic}]}} In[2]:= DensityPlot[Evaluate[u[t, x] /. %], {t, 0, 10}, {x, 0, 5}, PlotRange -> All, ColorFunction -> "SunsetColors"] Out[2]= [image] ``` --- Plot a saddle surface; the mesh curves show where the function is zero: ```wl In[1]:= DensityPlot[y ^ 2 - x ^ 2, {x, -4, 4}, {y, -4, 4}, MeshFunctions -> {#3&}, MeshStyle -> Directive[Dashed, Thickness[Medium]], Mesh -> {{0}}, PlotRange -> {-4, 4}, ClippingStyle -> Automatic] Out[1]= [image] ``` --- The 1, 2, 3, and $\infty$ norms, with the iso-norm mesh lines at 1/2, 1, and 3/2: ```wl In[1]:= Table[DensityPlot[Norm[{x, y}, p], {x, -1.5, 1.5}, {y, -1.5, 1.5}, Ticks -> None, MeshFunctions -> {#3&}, Mesh -> {{1 / 2, 1, 3 / 2}}, MeshStyle -> Directive[Dashed, GrayLevel[0.3]], Exclusions -> None], {p, {1, 2, 3, Infinity}}] Out[1]= {[image], [image], [image], [image]} ``` --- Show argument variation for sin, cos, tan, and cot over the complex plane: ```wl In[1]:= Table[DensityPlot[Arg[f[x + I y]], {x, -2Pi, 2Pi}, {y, -2, 2}, ColorFunction -> "SunsetColors", FrameTicks -> {Range[-2Pi, 2Pi, Pi], Automatic, None, None}], {f, {Sin, Cos, Tan, Cot}}] Out[1]= {[image], [image], [image], [image]} ``` --- Show the different complex components for a function: ```wl In[1]:= Table[DensityPlot[f[ArcSin[(x + I y) ^ 2]], {x, -2, 2}, {y, -2, 2}, ColorFunction -> "DarkRainbow", ExclusionsStyle -> Red], {f, {Re, Im, Abs, Arg}}] Out[1]= {[image], [image], [image], [image]} ``` --- Transform a function to expose more features: ```wl In[1]:= DensityPlot[(x - 1) ^ 2 + 100(x ^ 2 - y) ^ 2, {x, -2, 2}, {y, -2, 2}, ClippingStyle -> Automatic, ColorFunction -> "Rainbow", PlotPoints -> 30] Out[1]= [image] In[2]:= DensityPlot[Log[(x - 1) ^ 2 + 100(x ^ 2 - y) ^ 2], {x, -2, 2}, {y, -2, 2}, ClippingStyle -> Automatic, ColorFunction -> "Rainbow", PlotPoints -> 30] Out[2]= [image] ``` ### Properties & Relations (9) ``DensityPlot`` samples more points where it needs to: ```wl In[1]:= DensityPlot[Sin[x y], {x, 0, 3}, {y, 0, 3}, Mesh -> All] Out[1]= [image] ``` --- Use ``ContourPlot`` to get segmented iso curves and contour regions: ```wl In[1]:= ContourPlot[Sin[x y], {x, 0, 3}, {y, 0, 3}] Out[1]= [image] ``` --- Use ``ListDensityPlot`` for plotting continuous data: ```wl In[1]:= data = Table[Sin[x y], {x, 0, 3, .1}, {y, 0, 3, .1}]; In[2]:= ListDensityPlot[data] Out[2]= [image] ``` --- Use ``Plot3D`` to get 3D surfaces: ```wl In[1]:= Plot3D[Sin[x y], {x, 0, 3}, {y, 0, 3}] Out[1]= [image] ``` Add a ``ColorFunction`` to get an overlay density: ```wl In[2]:= Plot3D[Sin[x y], {x, 0, 3}, {y, 0, 3}, ColorFunction -> "SunsetColors", Mesh -> None] Out[2]= [image] ``` --- ``ComplexPlot`` plots the phase of a function using color and shades by the magnitude: ```wl In[1]:= ComplexPlot[Sin[z] ^ 3 / (z + 1) ^ 4, {z, -5 - 5I, 5 + 5I}] Out[1]= [image] In[2]:= DensityPlot[Arg[Sin[x + I * y] ^ 3 / (x + I * y + 1) ^ 4], {x, -5, 5}, {y, -5, 5}] Out[2]= [image] ``` --- Use ``ArrayPlot`` or ``MatrixPlot`` for discrete data: ```wl In[1]:= ArrayPlot[CellularAutomaton[30, {{1}, 0}, 30]] Out[1]= [image] In[2]:= MatrixPlot[Last@SchurDecomposition@RandomReal[1, {50, 50}]] Out[2]= [image] ``` --- Use ``Plot`` for univariate functions: ```wl In[1]:= Plot[Sin[x] + Sin[Sqrt[2]x], {x, 0, 20}] Out[1]= [image] ``` --- Use ``ParametricPlot`` for plane parametric curves and regions: ```wl In[1]:= {ParametricPlot[{Cos[θ], Sin[θ]}, {θ, 0, 2Pi}], ParametricPlot[{r Cos[θ], r Sin[θ]}, {θ, 0, 2Pi}, {r, 1 / 2, 1}]} Out[1]= [image] ``` --- Use ``ContourPlot3D`` and ``RegionPlot3D`` for implicit surfaces and regions: ```wl In[1]:= {ContourPlot3D[Sin[x y] == z, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}], RegionPlot3D[Sin[x y] ≥ z, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]} Out[1]= [image] ``` ### Possible Issues (2) With segmenting or piecewise color functions, the transition color borders may not be sharp: ```wl In[1]:= DensityPlot[Sin[y]Sin[y - x], {x, 0, 4Pi}, {y, 0, 4Pi}, ColorFunction -> (If[# > 0, Blue, White]&), ColorFunctionScaling -> False, PlotPoints -> 50] Out[1]= [image] ``` Use ``ContourPlot`` for segmenting problems instead: ```wl In[2]:= ContourPlot[Sin[y]Sin[y - x], {x, 0, 4Pi}, {y, 0, 4Pi}, Contours -> {0}, ContourShading -> {White, Blue}] Out[2]= [image] ``` --- Color functions or densities that change quickly may show artifacts: ```wl In[1]:= DensityPlot[Sin[x y], {x, 0, 4}, {y, 0, 4}, ColorFunction -> Hue] Out[1]= [image] ``` Use ``PlotPoints`` to increase the sampling density: ```wl In[2]:= DensityPlot[Sin[x y], {x, 0, 4}, {y, 0, 4}, ColorFunction -> Hue, PlotPoints -> 100] Out[2]= [image] ``` ### Neat Examples (2) Branch cuts for inverse trigonometric functions: ```wl In[1]:= $InverseTrigFunctions = {ArcSin, ArcCos, ArcSec, ArcCsc, ArcTan, ArcCot, ArcSinh, ArcCosh, ArcSech, ArcCsch, ArcTanh, ArcCoth}; In[2]:= Table[DensityPlot[Im[f[(x + I y) ^ 3]], {x, -2, 2}, {y, -2, 2}, ColorFunction -> "Pastel", ExclusionsStyle -> {None, Purple}, Mesh -> None, PlotLabel -> Im[f[(x + I y) ^ 3]], Ticks -> None], {f, $InverseTrigFunctions}] Out[2]= {[image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image]} ``` --- Real and imaginary part overlay mesh: ```wl In[1]:= $TrigFunctions = {Sin, Cos, Sec, Csc, Tan, Cot, ArcSin, ArcCos, ArcSec, ArcCsc, ArcTan, ArcCot, Sinh, Cosh, Sech, Csch, Tanh, Coth, ArcSinh, ArcCosh, ArcSech, ArcCsch, ArcTanh, ArcCoth}; In[2]:= Table[DensityPlot[Abs[f[x + I y]], {x, -2, 2}, {y, -2, 2}, MeshFunctions -> Function@@@{{{x, y, z}, Re[f[x + I y]]}, {{x, y, z}, Im[f[x + I y]]}}, MeshStyle -> {Opacity[1 / 3], Opacity[2 / 3]}, ColorFunction -> "NeonColors", PlotLabel -> f, Ticks -> None, Mesh -> 10], {f, $TrigFunctions}] Out[2]= {[image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image], [image]} ``` ## See Also * [`ListDensityPlot`](https://reference.wolfram.com/language/ref/ListDensityPlot.en.md) * [`DensityPlot3D`](https://reference.wolfram.com/language/ref/DensityPlot3D.en.md) * [`SliceDensityPlot3D`](https://reference.wolfram.com/language/ref/SliceDensityPlot3D.en.md) * [`ContourPlot`](https://reference.wolfram.com/language/ref/ContourPlot.en.md) * [`ArrayPlot`](https://reference.wolfram.com/language/ref/ArrayPlot.en.md) * [`Plot3D`](https://reference.wolfram.com/language/ref/Plot3D.en.md) * [`StreamPlot`](https://reference.wolfram.com/language/ref/StreamPlot.en.md) * [`VectorPlot`](https://reference.wolfram.com/language/ref/VectorPlot.en.md) * [`StreamDensityPlot`](https://reference.wolfram.com/language/ref/StreamDensityPlot.en.md) * [`VectorDensityPlot`](https://reference.wolfram.com/language/ref/VectorDensityPlot.en.md) * [`ColorFunction`](https://reference.wolfram.com/language/ref/ColorFunction.en.md) * [`RegionPlot`](https://reference.wolfram.com/language/ref/RegionPlot.en.md) * [`ComplexPlot`](https://reference.wolfram.com/language/ref/ComplexPlot.en.md) ## Tech Notes * [Density and Contour Plots](https://reference.wolfram.com/language/tutorial/TheStructureOfGraphicsAndSound.en.md#14270) ## Related Guides * [Function Visualization](https://reference.wolfram.com/language/guide/FunctionVisualization.en.md) * [Solvers over Regions](https://reference.wolfram.com/language/guide/GeometricSolvers.en.md) ## History * Introduced in 1988 (1.0) \| [Updated in 2007 (6.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn60.en.md) ▪ [2008 (7.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn70.en.md) ▪ [2012 (9.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn90.en.md) ▪ [2014 (10.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn100.en.md) ▪ [2017 (11.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn111.en.md) ▪ [2021 (13.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn130.en.md)