--- title: "ListVectorDensityPlot" language: "en" type: "Symbol" summary: "ListVectorDensityPlot[varr] generates a vector density plot from an array varr of vector and scalar field values {{vxij, vyij}, rij}. ListVectorDensityPlot[{{{x1, y1}, {{vx1, vy1}, r1}}, ...}] generates a vector density plot from vector and scalar field values {{vxi, vyi}, ri} given at specified points {xi, yi}. ListVectorDensityPlot[{data1, data2, ...}] plots data for several vector and scalar fields." keywords: - list vector density plot - plot vector field - vizualise vector field - arrow plot canonical_url: "https://reference.wolfram.com/language/ref/ListVectorDensityPlot.html" source: "Wolfram Language Documentation" related_guides: - title: "Vector Visualization" link: "https://reference.wolfram.com/language/guide/VectorVisualization.en.md" related_functions: - title: "ListVectorPlot" link: "https://reference.wolfram.com/language/ref/ListVectorPlot.en.md" - title: "ListVectorPlot3D" link: "https://reference.wolfram.com/language/ref/ListVectorPlot3D.en.md" - title: "ListStreamPlot3D" link: "https://reference.wolfram.com/language/ref/ListStreamPlot3D.en.md" - title: "ListStreamDensityPlot" link: "https://reference.wolfram.com/language/ref/ListStreamDensityPlot.en.md" - title: "ListLineIntegralConvolutionPlot" link: "https://reference.wolfram.com/language/ref/ListLineIntegralConvolutionPlot.en.md" - title: "VectorDensityPlot" link: "https://reference.wolfram.com/language/ref/VectorDensityPlot.en.md" - title: "ListDensityPlot" link: "https://reference.wolfram.com/language/ref/ListDensityPlot.en.md" - title: "ListContourPlot" link: "https://reference.wolfram.com/language/ref/ListContourPlot.en.md" --- # ListVectorDensityPlot ListVectorDensityPlot[varr] generates a vector density plot from an array varr of vector and scalar field values {{vxij, vyij}, rij}. ListVectorDensityPlot[{{{x1, y1}, {{vx1, vy1}, r1}}, …}] generates a vector density plot from vector and scalar field values {{vxi, vyi}, ri} given at specified points {xi, yi}. ListVectorDensityPlot[{data1, data2, …}] plots data for several vector and scalar fields. ## Details and Options * ``ListVectorDensityPlot`` generates a vector plot of the vector field, superimposed on a background density plot of the scalar field. * ``ListVectorDensityPlot`` displays a vector field by drawing arrows normalized to a fixed length. The arrows are colored by default according to the magnitude $\left\| \left\{v_x,v_y\right\}\right\|$ of the vector field. The vectors are drawn over a density plot of the scalar field ``r``. * The magnitude $\left\| \left\{v_x,v_y\right\}\right\|$ of the vector field is used for the scalar field ``r`` by default. * The plot visualizes the set $\left\{\left.\left\{\left\{v_x,v_y\right\},r\right\}\right|\{x,y\}\in \text{reg}\right\}$, where $\text{\textit{reg}}$ is the region for which $\left\{v_x,v_y\right\}$ is defined. [image] * ``ListVectorDensityPlot`` interpolates the data into vector function $\left\{v_x(x,y),v_y(x,y)\right\}$ and scalar function $r(x,y)$. * For regular data, the vector field $\left\{v_x,v_y\right\}$ has value ``varr[[i, j, 1]]`` and the scalar field has value ``varr[[i, j, 2]]`` at $\{x,y\}=\{i,j\}$. * For irregular data, the vector field $\left\{v_x,v_y\right\}$ has value ``{vxi, vyi}`` and the scalar field has value ``ri``at $\{x,y\}=\left\{\text{\textit{$x$}}_{\text{\textit{$i$}}},\text{\textit{$y$}}_{\text{\textit{$i$}}}\right\}$. * If no scalar field values are given, they are taken to be the norm of the vector field. * ``ListVectorDensityPlot[varr]`` arranges successive rows of ``varr`` up the page, and successive columns across. * ``ListVectorDensityPlot`` by default interpolates the data given and plots vectors for the vector field at a regular grid of positions. * With the setting ``VectorPoints -> All``, ``ListVectorDensityPlot`` instead shows vectors associated with the particular vector field data points given. * ``ListVectorDensityPlot`` 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 box ratios for simulated lighting | | ClippingStyle | Automatic | how to display arrows outside the vector range | | ColorFunction | Automatic | how to color background densities | | ColorFunctionScaling | True | whether to scale arguments to ColorFunction | | DataRange | Automatic | the range of x and y values to assume for data | | EvaluationMonitor | None | expression to evaluate at every function evaluation | | Frame | True | whether to draw a frame around the plot | | FrameTicks | Automatic | frame tick marks | | LightingAngle | None | effective angle for simulated lighting | | MaxRecursion | Automatic | the maximum number of recursive subdivisions allowed for the scalar field | | Mesh | None | how many mesh lines to draw in the background | | MeshFunctions | {#5&} | how to determine the placement of mesh lines | | MeshShading | None | how to shade regions between mesh lines | | MeshStyle | Automatic | the style of mesh lines | | Method | Automatic | methods to use for the plot | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotLegends | None | legends for color gradients | | PlotRange | {Full, Full} | range of x, y values to include | | PlotRangePadding | Automatic | how much to pad the range of values | | PlotTheme | \$PlotTheme | overall theme for the plot | | RegionFunction | (True&) | determine what region to include | | ScalingFunctions | None | how to scale individual coordinates | | VectorColorFunction | Automatic | how to color vectors | | VectorColorFunctionScaling | True | whether to scale the arguments to VectorColorFunction | | VectorMarkers | Automatic | shape to use for vectors | | VectorPoints | Automatic | the number or placement of vectors to plot | | VectorRange | Automatic | range of vector lengths to show | | VectorScaling | Automatic | how to scale the sizes of arrows | | VectorSizes | Automatic | sizes of displayed arrows | | VectorStyle | Automatic | how to draw vectors | | WorkingPrecision | MachinePrecision | precision to use in internal computations | * The arguments supplied to functions in ``MeshFunctions``, ``RegionFunction``, ``ColorFunction``, and ``VectorColorFunction`` are ``x``, ``y``, ``vx``, ``vy``, ``r``. * The default setting ``MeshFunctions -> {#5&}`` draws mesh lines for the scalar field ``r``. * Possible settings for ``ScalingFunctions`` are: {Subscript[``s``, ``x``], Subscript[``s``, ``y``]} scale ``x`` and ``y`` axes * Common built-in scaling functions ``s`` include: | | | | | ----------- | ------- | --------------------------------------------------- | | "Log" | [image] | log scale with automatic tick labeling | | "Log10" | [image] | base-10 log scale with powers of 10 for ticks | | "SignedLog" | [image] | log-like scale that includes 0 and negative numbers | | "Reverse" | [image] | reverse the coordinate direction | | "Infinite" | [image] | infinite scale | ### 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 box ratios for simulated lighting | | ClippingStyle | Automatic | how to display arrows outside the vector range | | ColorFunction | Automatic | how to color background densities | | ColorFunctionScaling | True | whether to scale arguments to ColorFunction | | ContentSelectable | Automatic | whether to allow contents to be selected | | CoordinatesToolOptions | Automatic | detailed behavior of the coordinates tool | | DataRange | Automatic | the range of x and y values to assume for data | | Epilog | {} | primitives rendered after the main plot | | EvaluationMonitor | None | expression to evaluate at every function evaluation | | 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 for simulated lighting | | MaxRecursion | Automatic | the maximum number of recursive subdivisions allowed for the scalar field | | Mesh | None | how many mesh lines to draw in the background | | MeshFunctions | {#5&} | how to determine the placement of mesh lines | | MeshShading | None | how to shade regions between mesh lines | | MeshStyle | Automatic | the style of mesh lines | | Method | Automatic | methods to use for the plot | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotLabel | None | an overall label for the plot | | PlotLegends | None | legends for color gradients | | PlotRange | {Full, Full} | range of x, y values to include | | PlotRangeClipping | False | 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&) | determine what region to include | | 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 | | VectorColorFunction | Automatic | how to color vectors | | VectorColorFunctionScaling | True | whether to scale the arguments to VectorColorFunction | | VectorMarkers | Automatic | shape to use for vectors | | VectorPoints | Automatic | the number or placement of vectors to plot | | VectorRange | Automatic | range of vector lengths to show | | VectorScaling | Automatic | how to scale the sizes of arrows | | VectorSizes | Automatic | sizes of displayed arrows | | VectorStyle | Automatic | how to draw vectors | | WorkingPrecision | MachinePrecision | precision to use in internal computations | --- ## Examples (121) ### Basic Examples (2) Plot the vector field with its norm as a background interpolated from a specified set of vectors: ```wl In[1]:= ListVectorDensityPlot[Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]] Out[1]= [image] ``` --- Plot the vector field from data specifying coordinates and vectors: ```wl In[1]:= data = Table[{{x, y}, {y, x - x ^ 3}}, {x, -1.5, 1.5, 0.2}, {y, -2, 2, 0.2}]; In[2]:= ListVectorDensityPlot[data] Out[2]= [image] ``` ### Scope (20) #### Sampling (9) Plot streamlines for a regular collection of vectors, and give a data range for the domain: ```wl In[1]:= ListVectorDensityPlot[Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], DataRange -> {{-3, 3}, {-3, 3}}] Out[1]= [image] ``` --- Plot an irregular collection of vectors: ```wl In[1]:= data = Table[{{x, y} = RandomReal[{-3, 3}, {2}], {y, -x}}, {200}]; In[2]:= ListVectorDensityPlot[data] Out[2]= [image] ``` Show all of the specified field vectors: ```wl In[3]:= ListVectorDensityPlot[data, VectorPoints -> All] Out[3]= [image] ``` Explicitly set the number of vectors in each direction: ```wl In[4]:= ListVectorDensityPlot[data, VectorPoints -> {4, 7}] Out[4]= [image] ``` --- Use an explicit scalar field: ```wl In[1]:= data = Table[{{x, y}, {{y, x - x ^ 3}, Abs[x]}}, {x, -1.5, 1.5, 0.2}, {y, -2, 2, 0.2}]; In[2]:= ListVectorDensityPlot[data] Out[2]= [image] ``` --- Use an explicit scalar field on an irregular collection of vectors: ```wl In[1]:= data = Table[{{x, y} = RandomReal[{-1.5, 1.5}, {2}], {{y, x - x ^ 3}, Abs[x]}}, {300}]; In[2]:= ListVectorDensityPlot[data, VectorPoints -> All] Out[2]= [image] ``` --- Plot two vector fields with background based on the norm of the first: ```wl In[1]:= data1 = Table[{{x, y} = RandomReal[{-3, 3}, {2}], {y, -x}}, {200}]; data2 = Table[{{x, y} = RandomReal[{-3, 3}, {2}], {y ^ 2, x - y}}, {150}]; In[2]:= ListVectorDensityPlot[{data1, data2}, VectorPoints -> 10] Out[2]= [image] ``` --- Specify a list of points for showing field vectors: ```wl In[1]:= ListVectorDensityPlot[Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], DataRange -> {{-3, 3}, {-3, 3}}, VectorPoints -> RandomReal[{-3, 3}, {300, 2}]] Out[1]= [image] ``` --- Plot a vector field with specified vector densities: ```wl In[1]:= data = Table[{y, -x}, {x, -2, 2, 0.2}, {y, -2, 2, 0.2}]; In[2]:= Table[ListVectorDensityPlot[data, PlotLabel -> p, VectorPoints -> p], {p, {5, Coarse, Automatic, Fine}}] Out[2]= [image] ``` --- Plot the vectors that go through a set of seed points: ```wl In[1]:= points = {{-1, -1}, {-1, 1}, {1, -1}, {1, 1}}; data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, DataRange -> {{-3, 3}, {-3, 3}}, VectorPoints -> points, VectorSizes -> 2, Epilog -> {Red, PointSize[Medium], Point[points]}] Out[2]= [image] ``` --- Plot vectors over a specified region: ```wl In[1]:= ListVectorDensityPlot[Table[{{x, y} = RandomReal[{-3, 3}, {2}], {y, -x}}, {200}], RegionFunction -> Function[{x, y, vx, vy, n}, x y < 1]] Out[1]= [image] ``` #### Presentation (11) Plot a vector field with arrows scaled according to their magnitudes: ```wl In[1]:= ListVectorDensityPlot[Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], VectorScaling -> Automatic] Out[1]= [image] ``` --- Use a single color for the arrows: ```wl In[1]:= ListVectorDensityPlot[Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], VectorColorFunction -> None] Out[1]= [image] ``` --- Plot a vector field with arrows of specified size: ```wl In[1]:= data = Table[{x, -y}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= Table[ListVectorDensityPlot[data, VectorColorFunction -> Hue, PlotLabel -> s, VectorSizes -> s], {s, {Small, Medium, Large, 0.5}}] Out[2]= {[image], [image], [image], [image]} ``` --- Vary the arrowhead size: ```wl In[1]:= data = Table[{Sin[x], Cos[y]}, {x, -Pi, Pi, 0.2}, {y, -Pi, Pi, 0.2}]; In[2]:= Table[ListVectorDensityPlot[data, PlotLabel -> s, VectorAspectRatio -> s], {s, {0.25, 0.5}}] Out[2]= [image] ``` --- Plot a vector field with the background and vectors colored according to the field magnitude: ```wl In[1]:= VectorDensityPlot[{y, -x}, {x, -3, 3}, {y, -3, 3}, ColorFunction -> "Pastel", VectorColorFunction -> Hue] Out[1]= [image] ``` --- Set the style for multiple vector fields: ```wl In[1]:= data1 = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; data2 = Table[{-(1 + x - y ^ 2), -1 - x ^ 2 + y}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[{data1, data2}, VectorPoints -> 8, VectorStyle -> {{Red, "Drop"}, {Blue, "Dart"}}, VectorColorFunction -> None] Out[2]= [image] ``` --- Apply a vector style: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, VectorPoints -> 8, VectorStyle -> Arrowheads[{{0.03, Automatic, Graphics[Circle[]]}}]] Out[2]= [image] ``` --- Use a named appearance to draw the vectors: ```wl In[1]:= data = Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, VectorMarkers -> "Drop"] Out[2]= [image] ``` Style the vectors as well: ```wl In[3]:= ListVectorDensityPlot[data, VectorMarkers -> "Drop", VectorColorFunction -> None, VectorStyle -> Red] Out[3]= [image] ``` --- Use a legend for the color gradients: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, VectorStyle -> Red, VectorColorFunction -> None, PlotLegends -> Automatic] Out[2]= [image] ``` --- Use a theme with detailed frame ticks and grid lines: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, PlotTheme -> "Detailed"] Out[2]= [image] ``` Use a theme with a high-contrast color scheme and edge-fading rectangles: ```wl In[3]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[4]:= ListVectorDensityPlot[data, PlotTheme -> "Marketing"] Out[4]= [image] ``` --- Reverse the direction of the ``y`` axis: ```wl In[1]:= ListVectorDensityPlot[Flatten[Table[{{x, y}, {x, y - x}}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], 1], ScalingFunctions -> {None, "Reverse"}] Out[1]= [image] ``` ### Options (87) #### Background (1) Use colored backgrounds: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], Background -> LightGray] Out[1]= [image] ``` #### BoundaryStyle (2) By default, region boundaries have no style: ```wl In[1]:= data = Table[{Sin[x], Cos[y]}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data] Out[2]= [image] ``` --- Change the style of the boundary: ```wl In[1]:= data = Table[{Sin[x], Cos[y]}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, BoundaryStyle -> Directive[Thick, Black]] Out[2]= [image] ``` #### ColorFunction (6) Color the field magnitude using ``Hue`` : ```wl In[1]:= ListVectorDensityPlot[Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], MaxRecursion -> 2, ColorFunction -> Hue] Out[1]= [image] ``` --- Color using ``Hue`` based on $\sin (x y)$ : ```wl In[1]:= ListVectorDensityPlot[Table[{{y, -x}, Sin[x y]}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], MaxRecursion -> 2, ColorFunction -> Hue] Out[1]= [image] ``` --- Use any named color gradient from ``ColorData``: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], ColorFunction -> "Pastel"] Out[1]= [image] ``` --- Use ``ColorData`` for predefined color gradients: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, ColorFunction -> Function[{x, y, vx, vy, n}, ColorData["SolarColors"][x y]]] Out[2]= [image] ``` --- Specify a color function that blends two colors by the $x$ coordinate: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, ColorFunction -> Function[{x, y, vx, vy, n}, Blend[{Green, Red}, x]]] Out[2]= [image] ``` --- Use ``ColorFunctionScaling -> False`` to get unscaled values: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, MaxRecursion -> 4, ColorFunctionScaling -> False, ColorFunction -> Function[{x, y, vx, vy, n}, ColorData[22][Round[n]]]] Out[2]= [image] ``` #### ColorFunctionScaling (4) By default, scaled values are used: ```wl In[1]:= data = Table[{y Cos[x y], x Cos[x y]}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, MaxRecursion -> 2, ColorFunction -> GrayLevel] Out[2]= [image] ``` --- Use ``ColorFunctionScaling -> False`` to get unscaled values: ```wl In[1]:= data = Table[{y Cos[x y], x Cos[x y]}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, MaxRecursion -> 4, ColorFunctionScaling -> False, ColorFunction -> Function[{x, y, vx, vy, n}, ColorData[22][Round[n]]]] Out[2]= [image] ``` --- Use unscaled coordinates in the $x$ direction and scaled coordinates in the $y$ direction: ```wl In[1]:= data = Table[{y Cos[x y], x Cos[x y]}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, ColorFunctionScaling -> {False, True}, ColorFunction -> Function[{x, y, vx, vy, n}, Hue[x, y, 1]]] Out[2]= [image] ``` --- Explicitly specify the scaling for each color function argument: ```wl In[1]:= data = Table[{y Cos[x y], x Cos[x y]}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, MaxRecursion -> 3, ColorFunctionScaling -> {True, True, True, True, False}, ColorFunction -> Function[{x, y, vx, vy, n}, Hue[n, y, 1]]] Out[2]= [image] ``` #### DataRange (1) By default, the data range is taken to be the index range of the data array: ```wl In[1]:= data = Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.3}]; Dimensions[data] Out[1]= {31, 21, 2} In[2]:= ListVectorDensityPlot[data] Out[2]= [image] ``` Specify the data range for the domain: ```wl In[3]:= ListVectorDensityPlot[data, DataRange -> {{-3, 3}, {-3, 3}}] Out[3]= [image] ``` #### EvaluationMonitor (1) Count the number of times the vector field function is evaluated: ```wl In[1]:= Block[{k = 0}, ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], EvaluationMonitor :> k++];k] Out[1]= 350 ``` #### MaxRecursion (1) Refine the plot where it changes quickly: ```wl In[1]:= data = Table[{{y, -x}, Sin[x y]}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= Table[ListVectorDensityPlot[data, PlotLabel -> r, ColorFunction -> Hue, MaxRecursion -> r], {r, 0, 2}] Out[2]= {[image], [image], [image]} ``` #### Mesh (5) By default, no mesh lines are displayed: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], Mesh -> None] Out[1]= [image] ``` --- Show the initial and final sampling meshes: ```wl In[1]:= Table[ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], PlotLabel -> m, Mesh -> m], {m, {Full, Automatic}}] Out[1]= [image] ``` --- Use a specific number of mesh lines: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], Mesh -> 5] Out[1]= [image] ``` --- Specify mesh lines: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], Mesh -> {{2, 4, 5}}] Out[1]= [image] ``` --- Use different styles for different mesh lines: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, Mesh -> {{{2, Thick}, {4, Yellow}, {5, {Thick, Red, Dashed}}}}] Out[2]= [image] ``` #### MeshFunctions (3) By default, mesh lines correspond to the magnitude of the field: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, Mesh -> 5, MeshFunctions -> Function[{x, y, vx, vy, n}, n]] Out[2]= [image] ``` --- Use the $x$ value as the mesh function: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, Mesh -> 5, MeshFunctions -> Function[{x, y, vx, vy, n}, x]] Out[2]= [image] ``` --- Use mesh lines corresponding to fixed distances from the origin: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, DataRange -> {{-3, 3}, {-3, 3}}, Mesh -> 5, MeshFunctions -> Function[{x, y, vx, vy, n}, Sqrt[x ^ 2 + y ^ 2]]] Out[2]= [image] ``` #### MeshShading (3) Use ``None`` to remove regions: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, Mesh -> 10, MeshShading -> {Red, None}] Out[2]= [image] ``` --- Styles are used cyclically: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, Mesh -> 10, MeshShading -> {Red, Yellow, Green, None}] Out[2]= [image] ``` --- Use indexed colors from ``ColorData`` cyclically: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, Mesh -> 10, MeshShading -> ColorData[51, "ColorList"]] Out[2]= [image] ``` #### MeshStyle (1) Apply a variety of styles to the mesh lines: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= Table[ListVectorDensityPlot[data, Mesh -> 10, MeshStyle -> s], {s, {Red, Thick, Directive[Red, Dashed]}}] Out[2]= [image] ``` #### PerformanceGoal (2) Generate a higher-quality plot: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], PerformanceGoal -> "Quality"] Out[1]= [image] ``` --- Emphasize performance, possibly at the cost of quality: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], PerformanceGoal -> "Speed"] Out[1]= [image] ``` #### PlotLegends (2) Use legends for vector color gradients: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -2, 2, .2}, {y, -2, 2, .2}], PlotLegends -> Automatic] Out[1]= [image] ``` --- Use ``Placed`` to put legends above the plot: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -2, 2, .2}, {y, -2, 2, .2}], PlotLegends -> Placed[Automatic, Above]] Out[1]= [image] ``` #### PlotRange (7) The full plot range is used by default: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], PlotRange -> Full] Out[1]= [image] ``` --- Specify an explicit limit for both $x$ and $y$ ranges: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], DataRange -> {{-3, 3}, {-3, 3}}, PlotRange -> 2] Out[1]= [image] ``` --- Specify an explicit $x$ range: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], DataRange -> {{-3, 3}, {-3, 3}}, PlotRange -> {{-1, 2}, Automatic}] Out[1]= [image] ``` --- Specify an explicit $x$ minimum range: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], DataRange -> {{-3, 3}, {-3, 3}}, PlotRange -> {{0, Full}, Automatic}] Out[1]= [image] ``` --- Specify an explicit $y$ range: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], DataRange -> {{-3, 3}, {-3, 3}}, PlotRange -> {Full, {0, 2}}] Out[1]= [image] ``` --- Specify an explicit $y$ maximum range: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], DataRange -> {{-3, 3}, {-3, 3}}, PlotRange -> {Full, {Full, 5}}] Out[1]= [image] ``` --- Specify different $x$ and $y$ ranges: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], DataRange -> {{-3, 3}, {-3, 3}}, PlotRange -> {{-2, 1.5}, {-1, 2}}] Out[1]= [image] ``` #### PlotTheme (2) Use a theme with simple ticks and grid lines in a high-contrast color scheme: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], DataRange -> {{-3, 3}, {-3, 3}}, PlotTheme -> "Business"] Out[1]= [image] ``` --- Change the color scheme: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], DataRange -> {{-3, 3}, {-3, 3}}, PlotTheme -> "Marketing"] Out[1]= [image] ``` #### RegionBoundaryStyle (4) The boundaries of full rectangular regions are not shown: ```wl In[1]:= data = Table[{Cos[x], y}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, DataRange -> {{-3, 3}, {-3, 3}}] Out[2]= [image] ``` --- Show the region being plotted: ```wl In[1]:= data = Table[{Cos[x], y}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, DataRange -> {{-3, 3}, {-3, 3}}, RegionFunction -> Function[{x, y, vx, vy, n}, x^2 + y^2 ≤ 9]] Out[2]= [image] ``` --- Specify a style for the boundary: ```wl In[1]:= data = Table[{Cos[x], y}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, DataRange -> {{-3, 3}, {-3, 3}}, RegionFunction -> Function[{x, y, vx, vy, n}, x^2 + y^2 ≤ 9], RegionBoundaryStyle -> Directive[Thick, Black]] Out[2]= [image] ``` --- Specify a style for full rectangular regions: ```wl In[1]:= data = Table[{Cos[x], y}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, DataRange -> {{-3, 3}, {-3, 3}}, RegionBoundaryStyle -> Directive[Thick, Black]] Out[2]= [image] ``` #### RegionFunction (3) Plot vectors only over certain quadrants: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, DataRange -> {{-3, 3}, {-3, 3}}, RegionFunction -> Function[{x, y, vx, vy, n}, x y < 1]] Out[2]= [image] ``` --- Plot vectors only over regions where the field magnitude is above a given threshold: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, DataRange -> {{-3, 3}, {-3, 3}}, RegionFunction -> Function[{x, y, vx, vy, n}, n > 6], MaxRecursion -> 1] Out[2]= [image] ``` --- Use any logical combination of conditions: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, DataRange -> {{-3, 3}, {-3, 3}}, RegionFunction -> Function[{x, y, vx, vy, n}, 4 < n < 9 || n < 2], MaxRecursion -> 2] Out[2]= [image] ``` #### ScalingFunctions (3) By default, linear scales are used: ```wl In[1]:= ListVectorDensityPlot[Table[{y, x - y}, {x, -1, 1, 0.1}, {y, -1, 1, 0.1}]] Out[1]= [image] ``` --- Use a log scale in the ``y`` direction: ```wl In[1]:= ListVectorDensityPlot[Table[{y, x - y}, {x, -1, 1, 0.1}, {y, -1, 1, 0.1}], ScalingFunctions -> {None, "Log"}] Out[1]= [image] ``` --- Reverse the direction of the ``y`` direction: ```wl In[1]:= ListVectorDensityPlot[Table[{y, x - y}, {x, -1, 1, 0.1}, {y, -1, 1, 0.1}], ScalingFunctions -> {None, "Reverse"}] Out[1]= [image] ``` #### VectorAspectRatio (1) Specify arrowhead size relative to the length of the arrow with ``VectorAspectRatio`` : ```wl In[1]:= ListVectorDensityPlot[Table[{Sin[x], Cos[y]}, {x, -Pi, Pi, 0.2}, {y, -Pi, Pi, 0.2}], VectorAspectRatio -> 1 / 2] Out[1]= [image] ``` #### VectorColorFunction (4) Color the vectors according to their norms: ```wl In[1]:= data = Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, VectorColorFunction -> Hue] Out[2]= [image] ``` --- Use any named color gradient from ``ColorData``: ```wl In[1]:= data = Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, VectorColorFunction -> "Rainbow"] Out[2]= [image] ``` --- Color the vectors according to their $x$ values: ```wl In[1]:= data = Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, VectorColorFunction -> Function[{x, y, vx, vy, n}, ColorData["ThermometerColors"][x]]] Out[2]= [image] ``` --- Use ``VectorColorFunctionScaling -> False`` to get unscaled values: ```wl In[1]:= data = Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, VectorColorFunction -> Function[{x, y, vx, vy, n}, ColorData[31][Round[n]]], VectorColorFunctionScaling -> False] Out[2]= [image] ``` #### VectorColorFunctionScaling (4) By default, scaled values are used: ```wl In[1]:= ListVectorDensityPlot[Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], VectorColorFunction -> Hue] Out[1]= [image] ``` --- Use ``VectorColorFunctionScaling -> False`` to get unscaled values: ```wl In[1]:= data = Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, VectorColorFunction -> Function[{x, y, vx, vy, n}, ColorData[31][Round[n]]], VectorColorFunctionScaling -> False] Out[2]= [image] ``` --- Use unscaled coordinates in the $x$ direction and scaled coordinates in the $y$ direction: ```wl In[1]:= data = Table[{y, -x}, {x, 0, 1, 0.1}, {y, 0, 1, 0.1}]; In[2]:= ListVectorDensityPlot[data, DataRange -> {{0, 2}, {0, 2}}, VectorColorFunction -> Function[{x, y, vx, vy, n}, Hue[x, y, 1]], VectorColorFunctionScaling -> {False, True}] Out[2]= [image] ``` --- Explicitly specify the scaling for each color function argument: ```wl In[1]:= data = Table[{y, -x}, {x, 0, 1, 0.1}, {y, 0, 1, 0.1}]; In[2]:= ListVectorDensityPlot[data, DataRange -> {{0, 2}, {0, 2}}, VectorColorFunction -> Function[{x, y, vx, vy, n}, Hue[n, y, 1]], VectorColorFunctionScaling -> {True, True, True, True, False}] Out[2]= [image] ``` #### VectorMarkers (10) Vectors are drawn as arrows by default: ```wl In[1]:= data = Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data] Out[2]= [image] ``` --- Use a named appearance to draw the vectors: ```wl In[1]:= data = Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, VectorMarkers -> "Drop"] Out[2]= [image] ``` --- Use different markers for different vector fields: ```wl In[1]:= data1 = Table[{{x, y}, {y, -x}}, {x, -5, 5}, {y, -5, 5}]; data2 = Table[{{x, y}, {x, y}}, {x, -5, 5}, {y, -5, 5}]; In[2]:= ListVectorDensityPlot[{data1, data2}, VectorMarkers -> {"Drop", "Dart"}] Out[2]= [image] ``` --- By default, markers are centered on vector points: ```wl In[1]:= data = Table[{{x, y}, {y, -x}}, {x, -5, 5}, {y, -5, 5}]; In[2]:= pts = Tuples[Range[-5, 5], 2]; In[3]:= ListVectorDensityPlot[data, VectorMarkers -> "Arrow", VectorPoints -> pts, Epilog -> {Red, Point[pts]}] Out[3]= [image] ``` Start the vectors at the points: ```wl In[4]:= ListVectorDensityPlot[data, VectorMarkers -> Placed["Arrow", "Start"], VectorPoints -> pts, Epilog -> {Red, Point[pts]}] Out[4]= [image] ``` End the vectors at the points: ```wl In[5]:= ListVectorDensityPlot[data, VectorMarkers -> Placed["Arrow", "End"], VectorPoints -> pts, Epilog -> {Red, Point[pts]}] Out[5]= [image] ``` --- Plot the vector field without arrowheads: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], VectorStyle -> "Segment"] Out[1]= [image] ``` --- Arrow vector styles: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= Table[ListVectorDensityPlot[data, VectorPoints -> 8, PlotLabel -> s, VectorStyle -> s], {s, {"Arrow", "DotArrow", "ArrowArrow"}}] Out[2]= {[image], [image], [image]} ``` --- Circular vector styles: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= Table[ListVectorDensityPlot[data, VectorPoints -> 10, PlotLabel -> s, VectorStyle -> s], {s, {"Circle", "Disk", "CircleArrow"}}] Out[2]= {[image], [image], [image]} ``` --- Dart vector styles: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= Table[ListVectorDensityPlot[data, VectorPoints -> 8, PlotLabel -> s, VectorStyle -> s], {s, {"Dart", "PinDart", "DoubleDart"}}] Out[2]= [image] ``` --- Vector styles with dots: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= Table[ListVectorDensityPlot[data, VectorPoints -> 8, PlotLabel -> s, VectorStyle -> s], {s, {"BarDot", "Dot", "DotArrow", "DotDot"}}] Out[2]= [image] ``` --- Pointer vector styles: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= Table[ListVectorDensityPlot[data, VectorPoints -> 8, PlotLabel -> s, VectorStyle -> s], {s, {"Drop", "Pointer", "BackwardPointer", "Toothpick"}}] Out[2]= [image] ``` #### VectorPoints (8) Use automatically determined vector points: ```wl In[1]:= ListVectorDensityPlot[Table[{y, -x}, {x, -1, 1, 0.2}, {y, -1, 1, 0.2}], VectorPoints -> Automatic] Out[1]= [image] ``` --- Show all of the specified field vectors: ```wl In[1]:= ListVectorDensityPlot[Table[{y, -x}, {x, -1, 1, 0.2}, {y, -1, 1, 0.2}], VectorPoints -> All] Out[1]= [image] ``` --- Use symbolic names to specify the set of field vectors: ```wl In[1]:= data = Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= Table[ListVectorDensityPlot[data, PlotLabel -> k, VectorPoints -> k], {k, {Fine, Coarse}}] Out[2]= [image] ``` --- Create a hexagonal grid of field vectors with the same number of arrows for $x$ and $y$ : ```wl In[1]:= ListVectorDensityPlot[Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], VectorPoints -> 5] Out[1]= [image] ``` --- Create a hexagonal grid of field vectors with a different number of arrows for $x$ and $y$ : ```wl In[1]:= ListVectorDensityPlot[Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], VectorPoints -> {4, 7}] Out[1]= [image] ``` --- Specify a list of points for showing field vectors: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[data, DataRange -> {{-3, 3}, {-3, 3}}, VectorPoints -> RandomReal[{-3, 3}, {300, 2}]] Out[2]= [image] ``` --- Use a different number of field vectors on a hexagonal grid: ```wl In[1]:= data = Table[{y ^ 2, x ^ 2 - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= Table[ListVectorDensityPlot[data, PlotLabel -> n, VectorPoints -> n, MaxRecursion -> 3], {n, 5, 20, 5}] Out[2]= [image] ``` --- The location for vectors is given in the middle of the drawn vector: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; points = {{-1, -1}, {-1, 1}, {1, -1}, {1, 1}}; In[2]:= ListVectorDensityPlot[data, DataRange -> {{-2, 2}, {-2, 2}}, VectorPoints -> points, VectorSizes -> 2, Epilog -> {Red, PointSize[Medium], Point[points]}] Out[2]= [image] ``` #### VectorScaling (2) Vectors by default are not scaled and have uniform length: ```wl In[1]:= ListVectorDensityPlot[Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], DataRange -> {{-3, 3}, {-3, 3}}] Out[1]= [image] ``` --- Use an automatically determined vector scale: ```wl In[1]:= ListVectorDensityPlot[Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], DataRange -> {{-3, 3}, {-3, 3}}, VectorScaling -> Automatic] Out[1]= [image] ``` #### VectorSizes (2) Specify the size of vectors: ```wl In[1]:= data = Table[{Sin[x], Cos[y]}, {x, -Pi, Pi, 0.2}, {y, -Pi, Pi, 0.2}]; In[2]:= Table[ListVectorDensityPlot[data, VectorPoints -> 10, PlotLabel -> s, VectorSizes -> s], {s, {0.5, 1}}] Out[2]= {[image], [image]} ``` --- Use symbolic names to control the size of vectors: ```wl In[1]:= data = Table[{y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= Table[ListVectorDensityPlot[data, VectorSizes -> s, PlotLabel -> s], {s, {Tiny, Small, Medium, Large}}] Out[2]= [image] ``` #### VectorStyle (5) Set the style for the displayed vectors: ```wl In[1]:= ListVectorDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}], VectorColorFunction -> None, VectorStyle -> Blue] Out[1]= [image] ``` --- Set the style for multiple vector fields: ```wl In[1]:= data1 = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; data2 = Table[{-(1 + x - y ^ 2), -1 - x ^ 2 + y}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[2]:= ListVectorDensityPlot[{data1, data2}, VectorColorFunction -> None, VectorStyle -> {Red, Blue}] Out[2]= [image] ``` --- Use ``Arrowheads`` to specify an explicit style of the arrowheads: ```wl In[1]:= data = Table[{x, y}, {x, -1, 1, 0.2}, {y, -1, 1, 0.2}]; In[2]:= ListVectorDensityPlot[data, VectorStyle -> Arrowheads[{{0.02, Automatic, Graphics[Circle[]]}}]] Out[2]= [image] ``` --- Specify both arrow tail and head: ```wl In[1]:= data = Table[{x, y}, {x, -1, 1, 0.2}, {y, -1, 1, 0.2}]; In[2]:= ListVectorDensityPlot[data, VectorPoints -> 9, VectorStyle -> Arrowheads[{{0.03, Automatic, Graphics[Rectangle[{-0.5, -0.5}, {0.5, 0.5}]]}, {0.03, Automatic, Graphics[Circle[]]}}]] Out[2]= [image] ``` --- Graphics primitives without ``Arrowheads`` are scaled based on the vector scale: ```wl In[1]:= data = Table[{x, y}, {x, -1, 1, 0.2}, {y, -1, 1, 0.2}]; In[2]:= ListVectorDensityPlot[data, VectorStyle -> Graphics[Circle[]]] Out[2]= [image] ``` ### Applications (2) Visualize a sampled vector field with the background based on the field's divergence: ```wl In[1]:= f = {Sin[x] ^ 3, -Cos[y] ^ 2}; df = Div[f, {x, y}]; data = Table[{f, df}, {x, -3, 3}, {y, -3, 3}]; In[2]:= ListVectorDensityPlot[data] Out[2]= [image] ``` --- Visualize the first horizontal and vertical Gaussian derivatives of an image: ```wl In[1]:= {dx, dy} = Table[ImageAdjust@GaussianFilter[[image], {6, 2}, δ], {δ, {{1, 0}, {0, 1}}}] Out[1]= [image] ``` Combine the vertical and horizontal Gaussian derivatives: ```wl In[2]:= data = Reverse /@ Transpose[2ImageData /@ {dx, dy} - 1, {3, 2, 1}]; In[3]:= ListVectorDensityPlot[data, VectorPoints -> 30, ColorFunction -> "Rainbow", VectorScaling -> Automatic, VectorAspectRatio -> 1 / 2] Out[3]= [image] ``` ### Properties & Relations (10) Use ``VectorDensityPlot`` to plot functions with a density plot of the scalar field: ```wl In[1]:= VectorDensityPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}] Out[1]= [image] ``` Use ``StreamDensityPlot`` to plot streamlines instead of vectors: ```wl In[2]:= StreamDensityPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}] Out[2]= [image] ``` --- Use ``ListVectorPlot`` for plotting data without a density plot: ```wl In[1]:= data = Table[{{x, y} = RandomReal[{-1.5, 1.5}, {2}], {{-1 - x ^ 2 + y, 1 + x - y ^ 2}, Abs[x + y]}}, {300}]; In[2]:= ListVectorPlot[data] Out[2]= [image] ``` Use ``ListStreamPlot`` to plot streamlines instead of vectors: ```wl In[3]:= ListStreamPlot[data] Out[3]= [image] ``` --- Use ``ListStreamDensityPlot`` to plot with streamlines instead of vectors : ```wl In[1]:= ListStreamDensityPlot[Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]] Out[1]= [image] ``` --- Use ``VectorPlot`` to plot functions without a density plot: ```wl In[1]:= VectorPlot[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3}, {y, -3, 3}] Out[1]= [image] ``` --- Use ``ListVectorDisplacementPlot`` to visualize the deformation of a region associated with a displacement vector field: ```wl In[1]:= data2D = Table[{{x, y} = RandomReal[{-1, 1}, {2}], {y, x - x^3}}, {300}]; In[2]:= data3D = Table[{{x, y, z} = RandomReal[{0, 1}, {3}], {y, x, -z}}, {300}]; In[3]:= ListVectorDisplacementPlot[data2D, Disk[], VectorPoints -> Automatic] Out[3]= [image] ``` Use ``ListVectorDisplacementPlot3D`` to visualize a deformation in 3D: ```wl In[4]:= ListVectorDisplacementPlot3D[data3D, Cuboid[], VectorPoints -> "Boundary"] Out[4]= [image] ``` --- Scalar fields can be plotted by themselves with ``ListDensityPlot`` : ```wl In[1]:= f = {-1 - x ^ 2 + y, 1 + x - y ^ 2}; s = Exp[Norm[f] / 10]; In[2]:= data = Table[{f, s}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; scalarData = Table[s, {x, -3, 3, 0.2}, {y, -3, 3, 0.2}]; In[3]:= {ListDensityPlot[scalarData, ColorFunction -> "Pastel"], ListVectorDensityPlot[data, ColorFunction -> "Pastel"]} Out[3]= {[image], [image]} ``` --- Use ``ListLineIntegralConvolutionPlot`` to plot the line integral convolution of vector field data: ```wl In[1]:= data = Table[{-1 - x ^ 2 + y, 1 + x - y ^ 2}, {x, -3, 3, .2}, {y, -3, 3, .2}]; In[2]:= ListLineIntegralConvolutionPlot[data] Out[2]= [image] ``` --- Use ``ListVectorPlot3D`` and ``ListStreamPlot3D`` to visualize 3D vector field data: ```wl In[1]:= data = Table[{z, -x, y}, {x, -1, 1, .1}, {y, -1, 1, .1}, {z, -1, 1, .1}]; In[2]:= {ListVectorPlot3D[data], ListStreamPlot3D[data]} Out[2]= {[image], [image]} ``` --- Plot vectors on surfaces with ``ListSliceVectorPlot3D`` : ```wl In[1]:= ListSliceVectorPlot3D[Table[{y, -x, z}, {z, -1, 1, .1}, {y, -1, 1, .1}, {x, -1, 1, .1}], Sphere[], DataRange -> {{-1, 1}, {-1, 1}, {-1, 1}}] Out[1]= [image] ``` --- Use ``GeoVectorPlot`` to plot vectors on a map: ```wl In[1]:= GeoVectorPlot[GeoVector[GeoPosition[CompressedData["«6488»"]] -> {{0.5706379232485146, Quantity[354.01686143622794, "AngularDegrees"]}, {0.48751317221734114, Quantity[327.18610724051325, "AngularDegrees"]}, {0.42407845478418604, Quantity[199.859344783 ... 904, "AngularDegrees"]}, {0.9366099079317551, Quantity[357.70482317743733, "AngularDegrees"]}, {0.8256817194791912, Quantity[242.07227508889662, "AngularDegrees"]}, {0.2661401145642981, Quantity[152.90003252279325, "AngularDegrees"]}}]] Out[1]= [image] ``` Use ``GeoStreamPlot`` to plot streamlines instead of vectors: ```wl In[2]:= GeoStreamPlot[GeoVector[GeoPosition[CompressedData["«6488»"]] -> {{0.5706379232485146, Quantity[354.01686143622794, "AngularDegrees"]}, {0.48751317221734114, Quantity[327.18610724051325, "AngularDegrees"]}, {0.42407845478418604, Quantity[199.859344783 ... 904, "AngularDegrees"]}, {0.9366099079317551, Quantity[357.70482317743733, "AngularDegrees"]}, {0.8256817194791912, Quantity[242.07227508889662, "AngularDegrees"]}, {0.2661401145642981, Quantity[152.90003252279325, "AngularDegrees"]}}]] Out[2]= [image] ``` ## See Also * [`ListVectorPlot`](https://reference.wolfram.com/language/ref/ListVectorPlot.en.md) * [`ListVectorPlot3D`](https://reference.wolfram.com/language/ref/ListVectorPlot3D.en.md) * [`ListStreamPlot3D`](https://reference.wolfram.com/language/ref/ListStreamPlot3D.en.md) * [`ListStreamDensityPlot`](https://reference.wolfram.com/language/ref/ListStreamDensityPlot.en.md) * [`ListLineIntegralConvolutionPlot`](https://reference.wolfram.com/language/ref/ListLineIntegralConvolutionPlot.en.md) * [`VectorDensityPlot`](https://reference.wolfram.com/language/ref/VectorDensityPlot.en.md) * [`ListDensityPlot`](https://reference.wolfram.com/language/ref/ListDensityPlot.en.md) * [`ListContourPlot`](https://reference.wolfram.com/language/ref/ListContourPlot.en.md) ## Related Guides * [Vector Visualization](https://reference.wolfram.com/language/guide/VectorVisualization.en.md) ## History * [Introduced in 2008 (7.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn70.en.md) \| [Updated in 2012 (9.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn90.en.md) ▪ [2014 (10.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn100.en.md) ▪ [2018 (11.3)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn113.en.md) ▪ [2020 (12.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn121.en.md) ▪ [2022 (13.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn131.en.md)