--- title: "PolarPlot" language: "en" type: "Symbol" summary: "PolarPlot[r, {\\[Theta], \\[Theta]min, \\[Theta]max}] generates a polar plot of a curve with radius r as a function of angle \\[Theta]. PolarPlot[{r1, r2, ...}, {\\[Theta], \\[Theta]min, \\[Theta]max}] makes a polar plot of curves with radius functions r1, r2, ...." keywords: - Archimedean spiral - Guilloche - logarithmic spiral - polar coordinates - spirals - POLAR_CONTOUR - polar - polarp - ezpolar - polar canonical_url: "https://reference.wolfram.com/language/ref/PolarPlot.html" source: "Wolfram Language Documentation" related_guides: - title: "Function Visualization" link: "https://reference.wolfram.com/language/guide/FunctionVisualization.en.md" - title: "Angles and Polar Coordinates" link: "https://reference.wolfram.com/language/guide/AnglesAndPolarCoordinates.en.md" related_functions: - title: "ListPolarPlot" link: "https://reference.wolfram.com/language/ref/ListPolarPlot.en.md" - title: "ParametricPlot" link: "https://reference.wolfram.com/language/ref/ParametricPlot.en.md" - title: "RevolutionPlot3D" link: "https://reference.wolfram.com/language/ref/RevolutionPlot3D.en.md" - title: "SphericalPlot3D" link: "https://reference.wolfram.com/language/ref/SphericalPlot3D.en.md" - title: "PieChart" link: "https://reference.wolfram.com/language/ref/PieChart.en.md" - title: "SectorChart" link: "https://reference.wolfram.com/language/ref/SectorChart.en.md" - title: "RadialAxisPlot" link: "https://reference.wolfram.com/language/ref/RadialAxisPlot.en.md" related_tutorials: - title: "Some Special Plots" link: "https://reference.wolfram.com/language/tutorial/GraphicsAndSound.en.md#31896" --- # PolarPlot PolarPlot[r, {θ, θmin, θmax}] generates a polar plot of a curve with radius r as a function of angle θ. PolarPlot[{r1, r2, …}, {θ, θmin, θmax}] makes a polar plot of curves with radius functions r1, r2, …. ## Details and Options * The angle $$\theta$$ is measured in radians, counterclockwise from the positive $x$ axis. * The $x$, $y$ position corresponding to $r$, $\theta$ is $r \cos (\theta )$, $r \sin (\theta )$. The value of $\theta$ need not be between $0$ and $2 \pi$. * ``PolarPlot`` treats the variable $$\theta$$ as local, effectively using ``Block``. * ``PolarPlot`` has attribute ``HoldAll`` and evaluates functions only after assigning specific numerical values to $$\theta$$. * In some cases, it may be more efficient to use ``Evaluate`` to evaluate functions symbolically before specific numerical values are assigned to $$\theta$$. * No curve is drawn in any region where a function evaluates to ``None``. * The following wrappers ``w`` can be used for ``r``: | | | | ----------------------- | ----------------------------------------------- | | Annotation[r, label] | provide an annotation for r | | Button[r, action] | evaluate action when the curve for r is clicked | | Callout[r, label] | label the function with a callout | | Callout[r, label, pos] | place the callout at relative position pos | | EventHandler[r, events] | define a general event handler for r | | Hyperlink[r, uri] | make the function a hyperlink | | Labeled[r, label] | label the function | | Labeled[r, label, pos] | place the label at relative position pos | | Legended[r, label] | identify the function in a legend | | PopupWindow[r, cont] | attach a popup window to the function | | StatusArea[r, label] | display in the status area on mouseover | | Style[r, styles] | show the function using the specified styles | | Tooltip[r, label] | attach a tooltip to the function | | Tooltip[r] | use functions as tooltips | * Wrappers ``w`` can be applied at multiple levels: | | | | -------------- | --------------------------- | | w[r] | wrap r | | w[{r1, r2, …}] | wrap a collection of curves | | w1[w2[…]] | use nested wrappers | * ``Callout``, ``Labeled`` and ``Placed`` can use the following positions ``pos``: | | | | --------------------------- | ---------------------------------------------------------- | | Automatic | automatically placed labels | | Above, Below, Before, After | positions around the curve | | u | near the curve at parameter u | | {x, y} | position near {x, y} | | Scaled[s] | scaled position s along the curve | | {s, Above}, {s, Below}, … | relative position at position s along the curve | | {pos, epos} | epos in label placed at relative position pos of the curve | * ``PolarPlot`` has the same options as ``Graphics``, with the following additions and changes: [] | | | | | :-------------------- | :---------------- | :---------------------------------------------------------- | | Axes | True | whether to draw axes | | AxesOrigin | {0, 0} | the origin where axes cross | | ColorFunction | Automatic | how to determine the coloring of curves | | ColorFunctionScaling | True | whether to scale arguments to ColorFunction | | EvaluationMonitor | None | expression to evaluate at every function evaluation | | Exclusions | Automatic | points in $\theta$ to exclude | | ExclusionsStyle | None | what to draw at excluded points | | LabelingSize | Automatic | maximum size of callouts and labels | | MaxRecursion | Automatic | the maximum number of recursive subdivisions allowed | | Mesh | None | how many mesh points to draw on each curve | | MeshFunctions | {#3&} | how to determine the placement of mesh points | | MeshShading | None | how to shade regions between mesh points | | MeshStyle | Automatic | the style for mesh points | | Method | Automatic | the method to use for refining curves | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotLabels | None | labels to use for curves | | PlotLegends | None | legends for curves | | PlotPoints | Automatic | initial number of sample points | | PlotRange | Automatic | the range of values to include | | PlotRangeClipping | True | whether to clip at the plot range | | PlotStyle | Automatic | graphics directives to specify the style for each curve | | PlotTheme | \$PlotTheme | overall theme for the plot | | PolarAxes | False | whether to draw polar axes | | PolarAxesOrigin | Automatic | where to draw polar axes | | PolarGridLines | None | polar gridlines to draw | | PolarTicks | Automatic | polar axes ticks | | 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 | | "Expressions" | use r1, r2, … as legend labels | | {lbl1, lbl2, …} | use lbl1, lbl2, … as legend labels | | Placed[lspec, …] | specify placement for legend | * ``ColorData["DefaultPlotColors"]`` gives the default sequence of colors used by ``PlotStyle``. * ``PolarPlot`` initially evaluates functions at a number of equally spaced sample points specified by ``PlotPoints``. Then it uses an adaptive algorithm to choose additional sample points, subdividing a given interval at most ``MaxRecursion`` times. * You should realize that with the finite number of sample points used, it is possible for ``PolarPlot`` to miss features in your function. To check your results, you should try increasing the settings for ``PlotPoints`` and ``MaxRecursion``. * ``On[PolarPlot::accbend]`` makes ``PolarPlot`` print a message if it is unable to reach a certain smoothness of curve. * With ``Mesh -> All``, ``PolarPlot`` will explicitly draw a point at every position on each curve where each function was sampled. * The arguments supplied to functions in ``MeshFunctions`` and ``RegionFunction`` are ``x``, ``y``, ``θ``, ``r``. Functions in ``ColorFunction`` are by default supplied with scaled versions of these arguments. * The functions are evaluated all along each curve. * With the default settings ``Exclusions -> Automatic`` and ``ExclusionsStyle -> None``, ``PolarPlot`` breaks curves at discontinuities it detects. ``Exclusions -> None`` joins across discontinuities. * ``PolarPlot`` normally returns ``Graphics[{Line[…], …}]``. * Possible settings for ``ScalingFunctions`` include: | | | | ---------------- | ------------------- | | sr | scale the r axis | | {sx, sy, sθ, sr} | scale x, y, θ and r | * 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 | * Scaling ``θ`` affects how the plot is sampled, but not the overall appearance of the plot. * In general, you cannot scale both ``x`` or ``y`` and ``r`` simultaneously. ### List of all options | | | | | ---------------------- | ----------------- | ---------------------------------------------------------------------------------- | | AlignmentPoint | Center | the default point in the graphic to align with | | AspectRatio | Automatic | ratio of height to width | | Axes | True | whether to draw axes | | AxesLabel | None | axes labels | | AxesOrigin | {0, 0} | the origin where axes 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 | | ColorFunction | Automatic | how to determine the coloring of curves | | 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 | | Epilog | {} | primitives rendered after the main plot | | EvaluationMonitor | None | expression to evaluate at every function evaluation | | Exclusions | Automatic | points in $\theta$ to exclude | | ExclusionsStyle | None | what to draw at excluded points | | FormatType | TraditionalForm | the default format type for text | | Frame | False | whether to put a frame around the plot | | FrameLabel | None | frame labels | | FrameStyle | {} | style specifications for the frame | | FrameTicks | Automatic | frame ticks | | 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 | | LabelingSize | Automatic | maximum size of callouts and labels | | LabelStyle | {} | style specifications for labels | | MaxRecursion | Automatic | the maximum number of recursive subdivisions allowed | | Mesh | None | how many mesh points to draw on each curve | | MeshFunctions | {#3&} | how to determine the placement of mesh points | | MeshShading | None | how to shade regions between mesh points | | MeshStyle | Automatic | the style for mesh points | | Method | Automatic | the method to use for refining curves | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotLabel | None | an overall label for the plot | | PlotLabels | None | labels to use for curves | | PlotLegends | None | legends for curves | | PlotPoints | Automatic | initial number of sample points | | PlotRange | Automatic | the range of 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 | | PlotStyle | Automatic | graphics directives to specify the style for each curve | | PlotTheme | \$PlotTheme | overall theme for the plot | | PolarAxes | False | whether to draw polar axes | | PolarAxesOrigin | Automatic | where to draw polar axes | | PolarGridLines | None | polar gridlines to draw | | PolarTicks | Automatic | polar axes ticks | | 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 (171) ### Basic Examples (3) Make a polar plot: ```wl In[1]:= PolarPlot[Sin[3t], {t, 0, Pi}] Out[1]= [image] ``` --- Make several polar plots: ```wl In[1]:= PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}] Out[1]= [image] ``` --- Style the curves: ```wl In[1]:= PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}, PlotStyle -> {Green, Directive[Dashed, Thick, Orange]}] Out[1]= [image] ``` ### Scope (23) #### Sampling (6) More points are sampled when the function changes quickly: ```wl In[1]:= PolarPlot[Sin[12θ], {θ, 0, 2Pi}] Out[1]= [image] ``` --- The plot range is selected automatically: ```wl In[1]:= PolarPlot[Tan[θ], {θ, 0, 2Pi}] Out[1]= [image] ``` --- Ranges where the function becomes nonreal are excluded: ```wl In[1]:= {PolarPlot[Sqrt[Sin[2θ]], {θ, 0, 2Pi}], PolarPlot[Im[Sqrt[Sin[2θ]]], {θ, 0, 2Pi}]} Out[1]= {[image], [image]} ``` --- The curve is split when there are discontinuities in the function: ```wl In[1]:= PolarPlot[Floor[θ], {θ, 0, 10}] Out[1]= [image] ``` --- Use ``PlotPoints`` and ``MaxRecursion`` to control adaptive sampling: ```wl In[1]:= Grid[Table[PolarPlot[1, {θ, 0, 2Pi}, PlotPoints -> pp, MaxRecursion -> mr], {mr, {0, 2}}, {pp, {5, 10}}]] Out[1]= | | | | ------- | ------- | | [image] | [image] | | [image] | [image] | ``` --- Use ``PlotRange`` to focus in on areas of interest: ```wl In[1]:= Table[PolarPlot[Exp[θ], {θ, 0, 10}, PlotRange -> pr], {pr, {10, 50, 250}}] Out[1]= [image] ``` #### Labeling and Legending (7) Use ``Callout`` to add the expressions as a label: ```wl In[1]:= PolarPlot[Callout[Sin[2t]], {t, 0, 2Pi}] Out[1]= [image] ``` Use any text as a label: ```wl In[2]:= PolarPlot[Callout[Sin[2t], "label"], {t, 0, 2Pi}] Out[2]= [image] ``` --- Place the label along the curve: ```wl In[1]:= Table[PolarPlot[Callout[Sin[3t], "label", i], {t, 0, 2Pi}, PlotLabel -> i], {i, {Pi / 4, 3Pi / 4}}] Out[1]= [image] ``` --- Place the label at a scaled position: ```wl In[1]:= Table[PolarPlot[Callout[Sin[3t], "label", Scaled[i]], {t, 0, 2Pi}, PlotLabel -> i], {i, {1 / 8, 3 / 8}}] Out[1]= [image] ``` --- Place the labels relative to the inside of the curve: ```wl In[1]:= Table[PolarPlot[Callout[Sin[3t], "label", pos], {t, 0, 2Pi}, PlotLabel -> pos], {pos, {Left, Right}}] Out[1]= [image] ``` Place the labels relative to the outside of the curve: ```wl In[2]:= Table[PolarPlot[Callout[Sin[3t], "label", pos], {t, 0, 2Pi}, PlotLabel -> pos], {pos, {Before, After}}] Out[2]= [image] ``` --- Label the curve with ``PlotLabels`` : ```wl In[1]:= PolarPlot[Cos[3t], {t, 0, 2Pi}, PlotLabels -> "Expressions"] Out[1]= [image] ``` Label multiple curves: ```wl In[2]:= PolarPlot[{Cos[t], Sin[t]}, {t, 0, Pi}, PlotLabels -> "Expressions"] Out[2]= [image] ``` --- Use a scaled position: ```wl In[1]:= PolarPlot[Callout[Sqrt[t], "label", Scaled[0.8]], {t, 0, 2Pi}] Out[1]= [image] ``` Specify the text position relative to the point: ```wl In[2]:= PolarPlot[Callout[Sqrt[t], "label", {Scaled[0.8], Below}], {t, 0, 2Pi}] Out[2]= [image] ``` --- Specify the label at ``{x, y}`` position: ```wl In[1]:= PolarPlot[Callout[Sqrt[t], "label", {1.5, -1}], {t, 0, 2Pi}] Out[1]= [image] ``` #### Presentation (10) Multiple curves are automatically colored to be distinct: ```wl In[1]:= PolarPlot[{1, 2, 3}, {θ, 0, 2Pi}] Out[1]= [image] ``` --- Provide explicit styling to different curves and regions: ```wl In[1]:= PolarPlot[{1, 2, 3}, {θ, 0, 2Pi}, PlotStyle -> {Blue, Directive[Red, Dashed], Directive[Purple, Thick]}] Out[1]= [image] ``` --- Add labels: ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, AxesLabel -> {x, y}, PlotLabel -> Sin[4θ]] Out[1]= [image] ``` --- Provide an interactive ``Tooltip`` for each curve: ```wl In[1]:= PolarPlot[{Tooltip[1], Tooltip[2], Tooltip[3]}, {θ, 0, 2Pi}] Out[1]= [image] ``` --- Create an overlay mesh: ```wl In[1]:= PolarPlot[θ, {θ, 0, 4Pi}, Mesh -> 20] Out[1]= [image] ``` --- Style the areas between mesh levels: ```wl In[1]:= PolarPlot[θ, {θ, 0, 4Pi}, PlotStyle -> Thick, Mesh -> 20, MeshShading -> {Red, Orange, Yellow}] Out[1]= [image] ``` --- Color by parameter values: ```wl In[1]:= PolarPlot[θ, {θ, 0, 4Pi}, PlotStyle -> Thick, ColorFunction -> Function[{x, y, θ}, Hue[θ / (2Pi)]], ColorFunctionScaling -> False] Out[1]= [image] ``` --- Legend multiple curves: ```wl In[1]:= PolarPlot[{1, 2, 3}, {θ, 0, 2Pi}, PlotStyle -> {Blue, Directive[Red, Dashed], Directive[Purple, Thick]}, PlotLegends -> "Expressions"] Out[1]= [image] ``` --- Use a theme with dark background and high-contrast colors: ```wl In[1]:= PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}, PlotTheme -> "Marketing"] Out[1]= [image] ``` Use a theme with minimal styling: ```wl In[2]:= PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}, PlotTheme -> "Minimal"] Out[2]= [image] ``` --- Use a log scale on the radial components of a polar plot: ```wl In[1]:= PolarPlot[θ, {θ, 0, 4Pi}, ScalingFunctions -> "Log"] Out[1]= [image] ``` Scale only the ``x`` direction: ```wl In[2]:= PolarPlot[θ, {θ, 0, 4Pi}, ScalingFunctions -> {"SignedLog", None, None, None}, AspectRatio -> 1] Out[2]= [image] ``` ### Options (134) #### AspectRatio (3) By default, the ratio of the height to width for the plot is determined automatically: ```wl In[1]:= PolarPlot[Sqrt[t], {t, 0, 2 π}] Out[1]= [image] ``` --- Make the height the same as the width with ``AspectRatio -> 1`` : ```wl In[1]:= PolarPlot[Sqrt[t], {t, 0, 2 π}, AspectRatio -> 1] Out[1]= [image] ``` --- ``AspectRatio -> Full`` adjusts the height and width to tightly fit inside other constructs: ```wl In[1]:= plot = PolarPlot[Sqrt[t], {t, 0, 2 π}, AspectRatio -> Full]; In[2]:= {Framed[Pane[plot, {50, 100}]], Framed[Pane[plot, {100, 100}]], Framed[Pane[plot, {100, 50}]]} Out[2]= [image] ``` #### Axes (4) By default, axes are drawn: ```wl In[1]:= PolarPlot[Sqrt[t], {t, 0, 2 π}] Out[1]= [image] ``` --- Use ``Axes -> False`` to turn off axes: ```wl In[1]:= PolarPlot[Sqrt[t], {t, 0, 2 π}, Axes -> False] Out[1]= [image] ``` --- Use ``AxesOrigin`` to specify where the axes intersect: ```wl In[1]:= PolarPlot[Sqrt[t], {t, 0, 2 π}, AxesOrigin -> {1, 1}] Out[1]= [image] ``` --- Turn each axis on individually: ```wl In[1]:= {PolarPlot[Sqrt[t], {t, 0, 2 π}, Axes -> {True, False}], PolarPlot[Sqrt[t], {t, 0, 2 π}, Axes -> {False, True}]} Out[1]= {[image], [image]} ``` #### AxesLabel (3) No axes labels are drawn by default: ```wl In[1]:= PolarPlot[Sqrt[t], {t, 0, 2 π}] Out[1]= [image] ``` --- Place a label on the $y$ axis: ```wl In[1]:= PolarPlot[Sqrt[t], {t, 0, 2 π}, AxesLabel -> r] Out[1]= [image] ``` --- Specify axes labels: ```wl In[1]:= PolarPlot[Sqrt[t], {t, 0, 2 π}, AxesLabel -> {r, r}] Out[1]= [image] ``` #### AxesOrigin (2) The position of the axes is determined automatically: ```wl In[1]:= PolarPlot[Sqrt[t], {t, 0, 2 π}] Out[1]= [image] ``` --- Specify an explicit origin for the axes: ```wl In[1]:= PolarPlot[Sqrt[t], {t, 0, 2 π}, AxesOrigin -> {1, 0}] Out[1]= [image] ``` #### AxesStyle (4) Change the style for the axes: ```wl In[1]:= PolarPlot[Sqrt[t], {t, 0, 2 π}, AxesStyle -> Red] Out[1]= [image] ``` --- Specify the style of each axis: ```wl In[1]:= PolarPlot[Sqrt[t], {t, 0, 2 π}, AxesStyle -> {Red, Blue}] Out[1]= [image] ``` --- Use different styles for the ticks and the axes: ```wl In[1]:= PolarPlot[Sqrt[t], {t, 0, 2 π}, AxesStyle -> Red, TicksStyle -> Black] Out[1]= [image] ``` --- Use different styles for the labels and the axes: ```wl In[1]:= PolarPlot[Sqrt[t], {t, 0, 2 π}, AxesStyle -> Green, LabelStyle -> Black] Out[1]= [image] ``` #### ColorFunction (4) Color the curve by scaled $x$, $y$, $\theta$, or $r$ value: ```wl In[1]:= Table[PolarPlot[Sin[4θ], {θ, 0, 2Pi}, ColorFunction -> f], {f, {Function[{x, y, t, r}, Hue[x]], Function[{x, y, t, r}, Hue[y]], Function[{x, y, t, r}, Hue[t]], Function[{x, y, t, r}, Hue[r]]}}] Out[1]= [image] ``` --- Use a named color gradient: ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, ColorFunction -> "DarkRainbow"] Out[1]= [image] ``` --- ``ColorFunction`` has higher priority than ``PlotStyle`` : ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, ColorFunction -> "DarkRainbow", PlotStyle -> Directive[Red, Thick]] Out[1]= [image] ``` --- Use red for the parameter $\theta >\pi$ : ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, ColorFunction -> Function[{x, y, t, r}, If[t > Pi, Red, Black]], ColorFunctionScaling -> False, PlotStyle -> Thick] Out[1]= [image] ``` #### ColorFunctionScaling (1) Color the curve by angle: ```wl In[1]:= PolarPlot[θ, {θ, 0, 4Pi}, PlotStyle -> Thick, ColorFunction -> Function[{x, y, t, r}, Hue[t / (2Pi)]], ColorFunctionScaling -> False] Out[1]= [image] ``` #### EvaluationMonitor (3) Find the list of parameter values evaluated: ```wl In[1]:= Reap[PolarPlot[Sin[4θ], {θ, 0, 2Pi}, EvaluationMonitor :> Sow[θ]];]//Short Out[1]//Short= {Null, {{0., 0.12333, 0.257035, «1772», 5.76497, 5.88713, 6.28103}}} ``` --- Find the coordinate values: ```wl In[1]:= data = Reap[PolarPlot[Sin[4θ], {θ, 0, 2Pi}, EvaluationMonitor :> Sow[Sin[4θ]{Cos[θ], Sin[θ]}]] ][[-1, 1]]; In[2]:= ListPlot[data ] Out[2]= [image] ``` --- Count how many times the function is evaluated: ```wl In[1]:= Block[{k = 0}, PolarPlot[Sin[4θ], {θ, 0, 2Pi}, EvaluationMonitor :> k++];k] Out[1]= 1778 ``` #### Exclusions (3) Automatically determine exclusions: ```wl In[1]:= PolarPlot[Floor[θ], {θ, 0, 2Pi}] Out[1]= [image] ``` --- Provide an explicit list of points for exclusions: ```wl In[1]:= PolarPlot[Tan[θ], {θ, 0, 2Pi}, Exclusions -> Range[Pi / 2, 2Pi, Pi], AspectRatio -> 1] Out[1]= [image] ``` --- Specify exclusions using equations: ```wl In[1]:= PolarPlot[Tan[θ], {θ, 0, 2Pi}, Exclusions -> {Cos[θ] == 0}, AspectRatio -> 1] Out[1]= [image] ``` #### ExclusionsStyle (2) Specify explicit styling for lines joining exclusion points: ```wl In[1]:= PolarPlot[Floor[θ], {θ, 0, 2Pi}, ExclusionsStyle -> Red] Out[1]= [image] ``` --- Provide styling for both exclusion points and the lines joining them: ```wl In[1]:= PolarPlot[Floor[θ], {θ, 0, 2Pi}, ExclusionsStyle -> {Dashed, PointSize[Medium]}] Out[1]= [image] ``` #### Frame (4) ``PolarPlot`` does not use a frame by default: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}] Out[1]= [image] ``` --- Use ``Frame -> True`` to turn on the frame: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> True] Out[1]= [image] ``` --- Draw a frame on the left and right edges: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> {{True, True}, {False, False}}] Out[1]= [image] ``` --- Draw a frame on the left and bottom edges: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> {{True, False}, {True, False}}] Out[1]= [image] ``` #### FrameLabel (3) Place a label along the bottom frame of a plot: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> True, FrameLabel -> "label"] Out[1]= [image] ``` --- Place labels on the bottom and left frame edges: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> True, FrameLabel -> {{Left, None}, {Bottom, None}}] Out[1]= [image] ``` --- Place labels on each of the edges in the frame: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> True, FrameLabel -> {{Left, Right}, {Bottom, Top}}] Out[1]= [image] ``` #### FrameStyle (2) Specify the style of the frame: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> True, FrameStyle -> Directive[Black, Thick]] Out[1]= [image] ``` --- Specify style for each frame edge: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> True, FrameStyle -> {{Directive[Black, Thick], Red}, {Directive[Gray, Thick], Directive[Blue]}}] Out[1]= [image] ``` #### FrameTicks (8) Frame ticks are placed automatically by default: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> True] Out[1]= [image] ``` --- Use a frame with no ticks: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> True, FrameTicks -> None] Out[1]= [image] ``` --- Use frame ticks on the bottom edge: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> True, FrameTicks -> {{None, None}, {Automatic, None}}] Out[1]= [image] ``` --- By default, the top and right edges have tick marks but no tick labels: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> True, FrameTicks -> Automatic] Out[1]= [image] ``` Use ``All`` to include tick labels on all edges: ```wl In[2]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> True, FrameTicks -> All] Out[2]= [image] ``` --- Place tick marks at specific positions: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> True, FrameTicks -> {{{-1, 0.5, 1}, Automatic}, {{-1, 1}, Automatic}}] Out[1]= [image] ``` --- Draw frame tick marks at the specified positions with specific labels: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> True, FrameTicks -> {{{{-1, -a}, {0.5, b}, {1, a}}, Automatic}, {{{-1, -a}, {1, a}}, Automatic}}] Out[1]= [image] ``` --- Specify the lengths for tick marks as a fraction of the graphics size: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Axes -> False, Frame -> True, FrameTicks -> {{{{-1, -a, .2}, {1, a, .2}}, Automatic}, {Automatic, Automatic}}] Out[1]= [image] ``` Use different sizes in the positive and negative directions for each tick mark: ```wl In[2]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0, .9 π}, Axes -> False, Frame -> True, FrameTicks -> {{{{-1, -a, {.1, .1}}, {0.5, b, {.2, .1}}, {1, a, {.2, .1}}}, Automatic}, {Automatic, Automatic}}] Out[2]= [image] ``` --- Specify a style for each frame tick: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0, .9 π}, Axes -> False, Frame -> True, FrameTicks -> {{{{-1, -a, {.1, .1}, Directive[Thick, Blue]}, {0.5, b, {.1, .1}, Directive[Thick, Darker@Green, Dashed]}, {1, a, {.1, .1}, Directive[Thick, Red]}}, Automatic}, {Automatic, Automatic}}] Out[1]= [image] ``` #### FrameTicksStyle (3) By default, the frame ticks and frame tick labels use the same styles as the frame: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0, .9 π}, Axes -> False, Frame -> True, FrameStyle -> Directive[Red]] Out[1]= [image] ``` --- Specify an overall style for the ticks, including the labels: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0, .9 π}, Axes -> False, Frame -> True, FrameTicksStyle -> Directive[Red, Thick]] Out[1]= [image] ``` --- Use different styles for the different frame edges: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0, .9 π}, Axes -> False, Frame -> True, FrameTicks -> All, FrameTicksStyle -> {{Directive[Black, Thick], Blue}, {Red, Darker@Green}}] Out[1]= [image] ``` #### ImageSize (7) Use named sizes such as ``Tiny``, ``Small``, ``Medium`` and ``Large`` : ```wl In[1]:= {PolarPlot[Sqrt[t], {t, 0, 2 π}, ImageSize -> Tiny], PolarPlot[Sqrt[t], {t, 0, 2 π}, ImageSize -> Small]} Out[1]= {[image], [image]} ``` --- Specify the width of the plot: ```wl In[1]:= {PolarPlot[Sqrt[t], {t, 0, 2 π}, ImageSize -> 150], PolarPlot[Sqrt[t], {t, 0, 2 π}, AspectRatio -> 1.5, ImageSize -> 150]} Out[1]= {[image], [image]} ``` Specify the height of the plot: ```wl In[2]:= {PolarPlot[Sqrt[t], {t, 0, 2 π}, ImageSize -> {Automatic, 150}], PolarPlot[Sqrt[t], {t, 0, 2 π}, AspectRatio -> 2, ImageSize -> {Automatic, 150}]} Out[2]= {[image], [image]} ``` --- Allow the width and height to be up to a certain size: ```wl In[1]:= {PolarPlot[Sqrt[t], {t, 0, 2 π}, ImageSize -> UpTo[200]], PolarPlot[Sqrt[t], {t, 0, 2 π}, 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]:= PolarPlot[Sqrt[t], {t, 0, 2 π}, ImageSize -> {200, 250}, Background -> LightBlue] Out[1]= [image] ``` Setting ``AspectRatio -> Full`` will fill the available space: ```wl In[2]:= PolarPlot[Sqrt[t], {t, 0, 2 π}, AspectRatio -> Full, ImageSize -> {200, 250}, Background -> LightBlue] Out[2]= [image] ``` --- Use maximum sizes for the width and height: ```wl In[1]:= {PolarPlot[Sqrt[t], {t, 0, 2 π}, ImageSize -> {UpTo[150], UpTo[100]}], PolarPlot[Sqrt[t], {t, 0, 2 π}, 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[PolarPlot[Sqrt[t], {t, 0, 2 π}, 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[PolarPlot[Sqrt[t], {t, 0, 2 π}, AspectRatio -> Full, ImageSize -> {Scaled[0.5], Scaled[0.5]}, Background -> LightBlue], {200, 100}]] Out[1]= [image] ``` #### LabelingSize (2) Textual labels are shown at their actual sizes: ```wl In[1]:= PolarPlot[Callout[Cos[Sin[t]], RandomWord[], {0.5, -0.5}], {t, 0, 2Pi}] Out[1]= [image] ``` Specify the size of the text: ```wl In[2]:= PolarPlot[Callout[Cos[Sin[t]], RandomWord[], {0.5, -0.5}], {t, 0, 2Pi}, LabelingSize -> 20] Out[2]= [image] ``` --- Image labels are resized to fit in the plot: ```wl In[1]:= PolarPlot[Callout[Cos[Sin[t]], [image], {0.5, -0.5}], {t, 0, 2Pi}] Out[1]= [image] ``` Specify the labeling size: ```wl In[2]:= PolarPlot[Callout[Cos[Sin[t]], [image], {0.5, -0.5}], {t, 0, 2Pi}, LabelingSize -> 60] Out[2]= [image] ``` #### MaxRecursion (1) Each level of ``MaxRecursion`` will adaptively subdivide the initial mesh into a finer mesh: ```wl In[1]:= Table[ PolarPlot[Sin[4θ], {θ, 0, 6Pi}, MaxRecursion -> i, Mesh -> All, Ticks -> None], {i, {0, 3, 6}}] Out[1]= [image] ``` #### Mesh (4) Show the initial and final sampling meshes: ```wl In[1]:= {PolarPlot[Sqrt[θ], {θ, 0, 2Pi}, Mesh -> Full], PolarPlot[Sqrt[θ], {θ, 0, 2Pi}, Mesh -> All]} Out[1]= [image] ``` --- Use 10 mesh points evenly spaced in the $\theta$ direction: ```wl In[1]:= PolarPlot[Sqrt[θ], {θ, 0, 2Pi}, Mesh -> 10] Out[1]= [image] ``` --- Use an explicit list of values for the mesh in the $\theta$ direction: ```wl In[1]:= PolarPlot[Sqrt[θ], {θ, 0, 2Pi}, Mesh -> {{Pi / 4, 5Pi / 4}}] Out[1]= [image] ``` --- Use explicit value and style for the $\theta$ mesh: ```wl In[1]:= PolarPlot[Sqrt[θ], {θ, 0, 2Pi}, Mesh -> {{{Pi / 4, PointSize[Large]}, {5Pi / 4, Red}}}] Out[1]= [image] ``` #### MeshFunctions (2) Use a mesh evenly spaced in the $x$, $y$, $\theta$, and $r$ directions: ```wl In[1]:= Table[PolarPlot[Sin[4θ], {θ, 0, 2Pi}, MeshFunctions -> {f}, Mesh -> 15], {f, {Function[{x, y, t, r}, x], Function[{x, y, t, r}, y], Function[{x, y, t, r}, t], Function[{x, y, t, r}, r]}}] Out[1]= [image] ``` --- Show five mesh levels in the $x$ direction (red) and 10 in the $y$ direction (blue): ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, Mesh -> {5, 10}, MeshFunctions -> {#1&, #2&}, MeshStyle -> {Directive[PointSize[Medium], Red], Blue}] Out[1]= [image] ``` #### MeshShading (6) Alternate red and blue arcs in the $u$ direction: ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, Mesh -> 15, MeshShading -> {Red, Blue}] Out[1]= [image] ``` --- Use ``None`` to remove segments: ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, Mesh -> 15, MeshShading -> {Red, None}] Out[1]= [image] ``` --- ``MeshShading`` can be used with ``PlotStyle`` : ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, PlotStyle -> Thick, Mesh -> 15, MeshShading -> {Red, Blue}] Out[1]= [image] ``` --- ``MeshShading`` has higher priority than ``PlotStyle`` for styling: ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, PlotStyle -> Green, Mesh -> 15, MeshShading -> {Red, Blue}] Out[1]= [image] ``` --- Use ``PlotStyle`` for some segments by setting ``MeshShading`` to ``Automatic`` : ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, PlotStyle -> Directive[Thick, Yellow], Mesh -> 15, MeshShading -> {Red, Automatic}] Out[1]= [image] ``` --- ``MeshShading`` can be used with ``ColorFunction`` : ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, PlotStyle -> Thick, ColorFunction -> Function[{x, y, θ, r}, Hue[θ]], Mesh -> 15, MeshShading -> {Black, Automatic}] Out[1]= [image] ``` #### MeshStyle (4) Automatically choose the mesh style: ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, Mesh -> 15] Out[1]= [image] ``` --- Use a red mesh in the $x$ direction: ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, Mesh -> 15, MeshFunctions -> {#1&}, MeshStyle -> Red] Out[1]= [image] ``` --- Use a red mesh in the $x$ direction and a blue mesh in the $y$ direction: ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, Mesh -> 15, MeshFunctions -> {#1&, #2&}, MeshStyle -> {Red, Blue}] Out[1]= [image] ``` --- Use big red mesh levels in the $\theta$ direction: ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, Mesh -> 15, MeshStyle -> Directive[PointSize[Large], Red]] Out[1]= [image] ``` #### PerformanceGoal (2) Generate a higher-quality plot: ```wl In[1]:= Timing[PolarPlot[Sin[8θ], {θ, 0, 2Pi}, PerformanceGoal -> "Quality"]] Out[1]= {0.040111, [image]} ``` --- Emphasize performance, possibly at the cost of quality: ```wl In[1]:= Timing[PolarPlot[Sin[8θ], {θ, 0, 2Pi}, PerformanceGoal -> "Speed"]] Out[1]= {0.024442, [image]} ``` #### PlotLabels (6) ``PlotLabels -> "Expressions"`` uses functions as curve labels: ```wl In[1]:= PolarPlot[{Sin[t], Cos[t]}, {t, 0, π}, PlotLabels -> "Expressions"] Out[1]= [image] ``` --- Specify the text to label the curves: ```wl In[1]:= PolarPlot[{Sin[t], Cos[t]}, {t, 0, π}, PlotLabels -> {"label1", "label2"}] Out[1]= [image] ``` --- Place the labels above the curves: ```wl In[1]:= PolarPlot[{Sin[t], Cos[t]}, {t, 0, π}, PlotLabels -> Placed[{"label1", "label2"}, Above]] Out[1]= [image] ``` Place the labels differently for each curve: ```wl In[2]:= PolarPlot[{Sin[t], Cos[t]}, {t, 0, π}, PlotLabels -> {Placed["label1", Above], Placed["label2", Below]}] Out[2]= [image] ``` --- Use callouts to identify the curves: ```wl In[1]:= PolarPlot[{Sin[t], Cos[t]}, {t, 0, π}, PlotLabels -> {Callout["label1", {Scaled[0.3], Above}], Callout["label2", {Scaled[0.75], Below}]}] Out[1]= [image] ``` --- Put labels relative to the outside of the curves: ```wl In[1]:= Table[PolarPlot[{Cos[t], Cos[3t]}, {t, 0, π}, PlotLabels -> Callout[{"label1", "label2"}, pos]], {pos, {After, Before}}] Out[1]= [image] ``` --- Use ``None`` to not add a label: ```wl In[1]:= PolarPlot[{Cos[t], Cos[3t]}, {t, 0, π}, PlotLabels -> {"label1", None}] Out[1]= [image] ``` #### PlotLegends (7) Use the automatic legend: ```wl In[1]:= PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}, PlotLegends -> Automatic] Out[1]= [image] ``` No legends are used by default: ```wl In[2]:= PolarPlot[Sin[3t], {t, 0, Pi}] Out[2]= [image] ``` Use legends: ```wl In[3]:= PolarPlot[Sin[3t], {t, 0, Pi}, PlotLegends -> All] Out[3]= [image] ``` Show no legends: ```wl In[4]:= PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}, PlotLegends -> None] Out[4]= [image] ``` --- ``PlotLegends`` automatically picks up ``PlotStyle`` option values: ```wl In[1]:= PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}, PlotStyle -> {Directive[Red, Dashed], Directive[Blue, AbsoluteThickness[5]]}, PlotLegends -> Automatic] Out[1]= [image] ``` Show expressions as legends in ``TraditionalForm`` : ```wl In[2]:= PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}, PlotLegends -> Placed["Expressions", Below]] Out[2]= [image] ``` --- Specify a list of labels for legends: ```wl In[1]:= PolarPlot[{1 + 1 / 10 Sin[3t], 1 + 1 / 10 Sin[6 t], 1 + 1 / 10 Sin[9t]}, {t, 0, 2π}, PlotStyle -> {Red, Blue, Orange}, PlotLegends -> {"ω = 3", "ω = 6", "ω = 9"}] Out[1]= [image] ``` --- Place legends outside: ```wl In[1]:= Table[PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}, PlotLegends -> Placed[Automatic, pl], PlotLabel -> pl], {pl, {After, Before}}] Out[1]= [image] In[2]:= Table[PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}, PlotLegends -> Placed[Automatic, pl], PlotLabel -> pl], {pl, {Above, Below}}] Out[2]= [image] ``` --- Place legends inside: ```wl In[1]:= Table[PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}, PlotLegends -> Placed[Automatic, pl], PlotLabel -> pl], {pl, {Right, Left}}] Out[1]= [image] In[2]:= Table[PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}, PlotLegends -> Placed[Automatic, pl], PlotLabel -> pl], {pl, {Top, Bottom}}] Out[2]= [image] ``` --- Legend layout changes automatically along with position: ```wl In[1]:= Table[PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}, PlotLegends -> Placed[Automatic, pl], PlotLabel -> pl], {pl, {After, Above}}] Out[1]= [image] ``` --- Use ``LineLegend`` to adjust the appearance of the legend: ```wl In[1]:= PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}, PlotLegends -> Placed[LineLegend["Expressions", LegendFunction -> Panel, LabelStyle -> {FontSize -> 12, Bold, FontFamily -> "Helvetica"}, LegendMargins -> 5], Below]] Out[1]= [image] ``` Specify ``LegendMarkers`` : ```wl In[2]:= marker = Graphics[BezierCurve[{{0, 0}, {1, 1}, {2, -1}, {3, 0}}], ImageSize -> Tiny] Out[2]= [image] In[3]:= PolarPlot[{1, 1 + 1 / 10 Sin[10t]}, {t, 0, 2Pi}, PlotStyle -> {Directive[Red, AbsoluteThickness[3], AbsoluteDashing[5]], Directive[Blue, AbsoluteThickness[2], AbsoluteDashing[8]]}, PlotLegends -> SwatchLegend["Expressions", LegendMarkers -> marker, LegendMarkerSize -> 40]] Out[3]= [image] ``` #### PlotPoints (1) Use more initial points to get a smoother plot: ```wl In[1]:= Table[PolarPlot[Sin[2θ], {θ, 0, 2Pi}, PlotPoints -> i, MaxRecursion -> 0], {i, {10, 15, 25, 50}}] Out[1]= {[image], [image], [image], [image]} ``` #### PlotRange (2) Show the curve where $-12\leq x\leq 12$ and $-12\leq y\leq 12$ : ```wl In[1]:= PolarPlot[θ, {θ, 0, 10}, PlotRange -> 12] Out[1]= [image] ``` --- With the natural range of values, the fine detail around the origin is not visible: ```wl In[1]:= PolarPlot[Exp[θ], {θ, 0, 10}] Out[1]= [image] ``` Use ``PlotRange`` to focus in on areas of interest: ```wl In[2]:= Table[PolarPlot[Exp[θ], {θ, 0, 10}, PlotRange -> pr], {pr, {10, 50, 250}}] Out[2]= [image] ``` #### PlotStyle (5) Use different style directives: ```wl In[1]:= Table[PolarPlot[Sin[4θ], {θ, 0, 2Pi}, PlotStyle -> ps], {ps, {Red, Thick, Dashed, Directive[Red, Thick]}}] Out[1]= [image] ``` --- By default, different styles are chosen for multiple curves and regions: ```wl In[1]:= PolarPlot[{θ, -θ}, {θ, 0, 4Pi}] Out[1]= [image] ``` --- Explicitly specify the style for different curves and regions: ```wl In[1]:= PolarPlot[{θ, -θ}, {θ, 0, 4Pi}, PlotStyle -> {Purple, Orange}] Out[1]= [image] ``` --- ``PlotStyle`` can be combined with ``ColorFunction`` : ```wl In[1]:= PolarPlot[Abs[Sin[4θ]], {θ, 0, 2Pi}, PlotStyle -> Thick, ColorFunction -> Function[{x, y, t, r}, Hue[r]]] Out[1]= [image] ``` --- ``PlotStyle`` can be combined with ``MeshShading`` : ```wl In[1]:= PolarPlot[Sin[4θ], {θ, 0, 2Pi}, PlotStyle -> Thick, Mesh -> 15, MeshShading -> {Red, Blue}] Out[1]= [image] ``` #### PlotTheme (1) Use a theme with polar grid lines and bright colors: ```wl In[1]:= PolarPlot[{θ, -θ}, {θ, 0, 4Pi}, PlotTheme -> "Business"] Out[1]= [image] ``` Add another theme with legends: ```wl In[2]:= PolarPlot[{θ, -θ}, {θ, 0, 4Pi}, PlotTheme -> {"Business", "Detailed"}] Out[2]= [image] ``` Change plot styles: ```wl In[3]:= PolarPlot[{θ, -θ}, {θ, 0, 4Pi}, PlotTheme -> {"Business", "Detailed"}, PlotStyle -> 96] Out[3]= [image] ``` #### PolarAxes (3) Use polar axes: ```wl In[1]:= PolarPlot[1 - 2Sin[θ], {θ, 0, 2π}, PlotRangePadding -> 3, PolarAxes -> True] Out[1]= [image] ``` --- Add polar grid lines: ```wl In[1]:= PolarPlot[1 - 2Sin[θ], {θ, 0, 2π}, PlotRangePadding -> 3, PolarAxes -> True, PolarGridLines -> True] Out[1]= [image] ``` --- Turn on the polar axis but not the radial axis: ```wl In[1]:= PolarPlot[1 - 2Sin[θ], {θ, 0, 2π}, PlotRangePadding -> 3, PolarAxes -> {True, False}, PolarGridLines -> True] Out[1]= [image] ``` #### PolarAxesOrigin (2) Reorient the polar and radial axes so that they intersect where $\theta =0$ and $r=3$ : ```wl In[1]:= PolarPlot[{2Sin[2θ], Cos[2θ]}, {θ, 0, 2π}, PlotRangePadding -> 2, PolarAxes -> True, PolarAxesOrigin -> {0, 3}] Out[1]= [image] ``` --- Turn off the polar ticks and move the radial axis below the plot: ```wl In[1]:= PolarPlot[{2Sin[2θ], Cos[2θ]}, {θ, 0, 2π}, PlotRangePadding -> 0.5, PolarAxes -> True, PolarAxesOrigin -> {{Bottom, Right}, Automatic}, PolarTicks -> {None, Automatic}] Out[1]= [image] ``` #### PolarGridLines (3) Use automatic polar grid lines: ```wl In[1]:= PolarPlot[3Cos[3θ], {θ, 0, 2Pi}, PlotRangePadding -> 3, PolarAxes -> True, PolarGridLines -> Automatic] Out[1]= [image] ``` --- Specify particular polar grid lines and ticks: ```wl In[1]:= PolarPlot[3Cos[3θ], {θ, 0, 2Pi}, PlotRangePadding -> 2, PolarAxes -> True, PolarGridLines -> {{0, (2π/3), (4π/3)}, Automatic}, PolarTicks -> {{0, (2π/3), (4π/3)}, Automatic}] Out[1]= [image] ``` --- Change the style of the specified grid lines: ```wl In[1]:= PolarPlot[3Cos[3θ], {θ, 0, 2Pi}, GridLinesStyle -> Orange, PlotRangePadding -> 2, PolarAxes -> True, PolarGridLines -> {{0, (2π/3), (4π/3)}, Automatic}, PolarTicks -> {{0, (2π/3), (4π/3)}, Automatic}] Out[1]= [image] ``` #### PolarTicks (3) Use automatic polar ticks: ```wl In[1]:= PolarPlot[1 - 2Sin[3θ], {θ, 0, 2Pi}, PlotRangePadding -> 4, PolarAxes -> True, PolarTicks -> Automatic] Out[1]= [image] ``` --- Change the labels for some of the ticks so the angle runs from $-\pi$ to $\pi$ instead of $0$ to $2 \pi$ : ```wl In[1]:= PolarPlot[1 - 2Sin[3θ], {θ, 0, 2Pi}, PlotRangePadding -> 4, PolarAxes -> True, PolarTicks -> {{0, (π/4), (π/2), (3 π/4), π, {(5 π/4), -(3π/4)}, {(3 π/2), -(π/2)}, {(7 π/4), -(π/4)}}, Automatic}] Out[1]= [image] ``` --- Display polar ticks in degrees and specify the radial ticks: ```wl In[1]:= PolarPlot[1 - 3Sin[5θ], {θ, 0, 2Pi}, PlotRangePadding -> 2, PolarAxes -> True, PolarTicks -> {"Degrees", Range[0, 5]}] Out[1]= [image] ``` #### RegionFunction (1) Show the plot where $r >\frac{1}{2}$ : ```wl In[1]:= PolarPlot[Abs[Sin[2θ]], {θ, 0, 2Pi}, RegionFunction -> Function[{x, y, t, r}, r > 0.5], PlotStyle -> Thick] Out[1]= [image] ``` #### ScalingFunctions (8) By default, ``PolarPlot`` uses natural scale in both axes: ```wl In[1]:= PolarPlot[5 Sin[4 θ] Cos[3 θ], {θ, 0, .9π}] Out[1]= [image] ``` --- Log-scaled plots will only plot intervals over which ``Log`` is defined: ```wl In[1]:= PolarPlot[5 Sin[4 θ] Cos[3 θ], {θ, 0, .9π}, ScalingFunctions -> {None, "Log", None, None}, AspectRatio -> 1] Out[1]= [image] ``` --- Use a shifted log scale to show a function with negative values: ```wl In[1]:= PolarPlot[5Sin[4 θ] Cos[3 θ], {θ, 0, .9π}, ScalingFunctions -> {None, "SignedLog", None, None}, AspectRatio -> 1] Out[1]= [image] ``` --- Use ``ScalingFunctions`` to reverse the coordinate direction in the $y$ direction: ```wl In[1]:= PolarPlot[5Sin[4 θ] Cos[3 θ], {θ, 0, .9π}, ScalingFunctions -> {None, "Reverse", None, None}] Out[1]= [image] ``` --- Use a reciprocal scale in the $y$ direction: ```wl In[1]:= PolarPlot[5 Sin[4 θ] Cos[3 θ], {θ, 0, .9π}, ScalingFunctions -> {Automatic, "Reciprocal", None, None}, AspectRatio -> 1] Out[1]= [image] ``` --- Use a scale defined by a function and its inverse: ```wl In[1]:= PolarPlot[Sin[4 θ] Cos[3 θ], {θ, 0, .9π}, ScalingFunctions -> {Automatic, {Log[#]&, Exp[-#]&}, None, None}, AspectRatio -> 1] Out[1]= [image] ``` --- Use a scale for ``r`` : ```wl In[1]:= {PolarPlot[5 Sin[4 θ] Cos[3 θ], {θ, 0, .9π}, ScalingFunctions -> "SignedLog"], PolarPlot[5 Sin[4 θ] Cos[3 θ], {θ, 0, .9π}, ScalingFunctions -> {Automatic, Automatic, Automatic, "SignedLog"}]} Out[1]= {[image], [image]} ``` --- Use a scale for ``θ``; it affects how the plot is sampled, but not the overall appearance of the plot: ```wl In[1]:= PolarPlot[5 Sin[4 θ] Cos[3 θ], {θ, 0, 6}, Mesh -> 20, MeshStyle -> Red] Out[1]= [image] In[2]:= PolarPlot[5 Sin[4 θ] Cos[3 θ], {θ, 0, 6}, ScalingFunctions -> {Automatic, Automatic, "Log", Automatic}, Mesh -> 20, MeshStyle -> Red] Out[2]= [image] ``` #### Ticks (9) Ticks are placed automatically in each plot: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}] Out[1]= [image] ``` --- Use ``Ticks -> None`` to draw axes without any tick marks: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Ticks -> None] Out[1]= [image] ``` --- Use ticks on the $x$ axis, but not the $y$ axis: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Ticks -> {Automatic, None}] Out[1]= [image] ``` --- Place tick marks at specific positions: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Ticks -> {{-1, 0.5, 1}, {-1, .5, 1}}] Out[1]= [image] ``` --- Draw tick marks at the specified positions with specific labels: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Ticks -> {{{-1, -a}, {0.5, b}, {1, a}}, {{-1, -a}, {1, a}}}] Out[1]= [image] ``` --- Use specific ticks on one axis and automatic ticks on the other: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Ticks -> {{{-1, -a}, {0.5, b}, {1, a}}, Automatic}] Out[1]= [image] ``` --- Specify the lengths for ticks as a fraction on graphics size: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Ticks -> {{{-1, -a, .4}, {0.5, b, .15}, {1, a, .2}}, Automatic}] Out[1]= [image] ``` Use different sizes in the positive and negative directions for each tick: ```wl In[2]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Ticks -> {{{-1, -a, {.4, .4}}, {0.5, b, {.15, .15}}, {1, a, {.2, .2}}}, Automatic}] Out[2]= [image] ``` --- Specify a style for each tick: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Ticks -> {{{-1, -a, {.4, .4}, Directive[Thick, Darker@Green]}, {0.5, b, {.15, .15}, Directive[Thick, Red]}, {1, a, {.2, .2}, Directive[Thick, Blue, Dashed]}}, Automatic}] Out[1]= [image] ``` --- Construct a function that places ticks at the midpoint and extremes of the axis: ```wl In[1]:= minMeanMax[min_, max_] := {{min, min}, {(max + min) / 2, (max + min) / 2}, {max, max}} In[2]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, Ticks -> {minMeanMax, minMeanMax}, PlotRangePadding -> None] Out[2]= [image] ``` #### TicksStyle (4) By default, the ticks and tick labels use the same styles as the axis: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, AxesStyle -> Red] Out[1]= [image] ``` --- Specify an overall ticks style, including the tick labels: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, TicksStyle -> Red] Out[1]= [image] ``` --- Specify ticks style for each of the axes: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, TicksStyle -> {Directive[Blue, Thick], Directive[Red]}] Out[1]= [image] ``` --- Use a different style for the tick labels and tick marks: ```wl In[1]:= PolarPlot[5Sin[2 θ] Cos[ 2θ], {θ, 0 , π}, TicksStyle -> Directive[Red, Thick], LabelStyle -> Blue] Out[1]= [image] ``` #### WorkingPrecision (2) Evaluate functions using machine-precision arithmetic: ```wl In[1]:= PolarPlot[Sin[2θ + 10 ^ 16], {θ, 0, 2Pi}, WorkingPrecision -> MachinePrecision] Out[1]= [image] ``` --- Evaluate functions using arbitrary-precision arithmetic: ```wl In[1]:= PolarPlot[Sin[2θ + 10 ^ 16], {θ, 0, 2Pi}, WorkingPrecision -> 20] Out[1]= [image] ``` ### Applications (4) Plot a circle: ```wl In[1]:= PolarPlot[1, {θ, 0, 2Pi}] Out[1]= [image] ``` A spiral: ```wl In[2]:= PolarPlot[θ, {θ, 0, 2Pi}] Out[2]= [image] ``` An oscillation around a circle: ```wl In[3]:= PolarPlot[1 + 1 / 10 Sin[10θ], {θ, 0, 2Pi}] Out[3]= [image] ``` --- Archimedean spirals of the form $r=a \theta ^{1/n}$ : ```wl In[1]:= {PolarPlot[θ ^ (1 / 2), {θ, 0, 20}, Frame -> True, PlotLabel -> r == θ ^ (1 / 2)], PolarPlot[θ, {θ, 0, 20}, Frame -> True, PlotLabel -> r == θ]} Out[1]= {[image], [image]} ``` Archimedean spirals of the form $r=a \left/ \theta ^{1/n}\right.$ : ```wl In[2]:= {PolarPlot[θ ^ -1, {θ, 0.1, 25}, PlotRange -> {{-1 / 2, 2.5}, All}, Frame -> True, PlotLabel -> r == θ ^ -1], PolarPlot[θ ^ (-1 / 2), {θ, 0.1, 25}, PlotRange -> {{-3 / 5, 2.5}, All}, Frame -> True, PlotLabel -> r == θ ^ (-1 / 2)]} Out[2]= {[image], [image]} ``` --- Logarithmic spirals have the form $r=a e^{b \theta }$ : ```wl In[1]:= PolarPlot[E ^ (θ / 10), {θ, 0, 30}, Frame -> True, PlotLabel -> r == E ^ (θ / 10)] Out[1]= [image] ``` --- Create a Guilloché pattern [[more info]](http://mathworld.wolfram.com/GuillochePattern.html) : ```wl In[1]:= guilloche[{a_, b_, c_, d_, e_, f_}, o : OptionsPattern[]] := PolarPlot[Evaluate[Flatten[{Table[(c + Sin[a x + d]) + ((b + Sin[b x + e]) - (c + Sin[a x + d])) (f + Sin[a x + n / Pi]) / 2, {n, 0, 19}]}]], {x, 0, 2 Pi}, o, Axes -> None, Frame -> False] In[2]:= guilloche[{4, 8, 20, 4.7, 1.8, 1}, PlotStyle -> Darker[Green, 0.5]] Out[2]= [image] ``` ### Properties & Relations (5) ``PolarPlot`` is a special case of ``ParametricPlot`` for curves: ```wl In[1]:= {PolarPlot[θ, {θ, 0, 4Pi}], ParametricPlot[{θ Cos[θ], θ Sin[θ]}, {θ, 0, 4Pi}]} Out[1]= {[image], [image]} ``` --- Use ``ListPolarPlot`` for data: ```wl In[1]:= {ListPolarPlot[Table[θ, {θ, 0, 2Pi, 0.1}]], PolarPlot[θ, {θ, 0, 2Pi}]} Out[1]= {[image], [image]} ``` --- Use ``Plot3D`` and ``ParametricPlot3D`` for function and parametric surfaces: ```wl In[1]:= Plot3D[(x ^ 2 + y ^ 2)Exp[-(x ^ 2 + y ^ 2)], {x, -2, 2}, {y, -2, 2}] Out[1]= [image] In[2]:= ParametricPlot3D[{-2 Cos[u] Cos[v] ^ 3, -2 Cos[v] ^ 2 Sin[u], 2 Tan[v]}, {u, 0, 2Pi}, {v, -1, 1}] Out[2]= [image] ``` --- Use ``RevolutionPlot3D`` and ``SphericalPlot3D`` for cylindrical and spherical coordinates: ```wl In[1]:= RevolutionPlot3D[-(ϕ + Sin[r]), {r, 0, 4Pi}, {ϕ, 0, 2Pi}] Out[1]= [image] In[2]:= SphericalPlot3D[1 + 1 / 5Sin[3v], {u, 0, Pi}, {v, 0, 2Pi}] Out[2]= [image] ``` --- Use ``ContourPlot`` and ``RegionPlot`` for implicit curves and regions: ```wl In[1]:= {ContourPlot[x ^ 2 + y ^ 2 == 4, {x, -2, 2}, {y, -2, 2}], RegionPlot[1 < x ^ 2 + y ^ 2 < 4, {x, -2, 2}, {y, -2, 2}]} Out[1]= {[image], [image]} ``` ### Neat Examples (2) ```wl In[1]:= PolarPlot[Exp[Cos[θ]] - 2Cos[4θ] + Sin[θ / 12]^5, {θ, 0, 20π}] Out[1]= [image] ``` --- ```wl In[1]:= PolarPlot[Evaluate[Table[Abs[Sin[θ + i]], {i, 0, 2Pi, 2Pi / 16}]], {θ, 0, 2Pi}, PlotStyle -> Thick, ColorFunction -> Function[{x, y, t, r}, Hue[r]], Axes -> False, RegionFunction -> Function[{x, y, t, r}, r < 0.555], ColorFunctionScaling -> False, PlotPoints -> 20, MaxRecursion -> 3] Out[1]= [image] ``` ## See Also * [`ListPolarPlot`](https://reference.wolfram.com/language/ref/ListPolarPlot.en.md) * [`ParametricPlot`](https://reference.wolfram.com/language/ref/ParametricPlot.en.md) * [`RevolutionPlot3D`](https://reference.wolfram.com/language/ref/RevolutionPlot3D.en.md) * [`SphericalPlot3D`](https://reference.wolfram.com/language/ref/SphericalPlot3D.en.md) * [`PieChart`](https://reference.wolfram.com/language/ref/PieChart.en.md) * [`SectorChart`](https://reference.wolfram.com/language/ref/SectorChart.en.md) * [`RadialAxisPlot`](https://reference.wolfram.com/language/ref/RadialAxisPlot.en.md) ## Tech Notes * [Some Special Plots](https://reference.wolfram.com/language/tutorial/GraphicsAndSound.en.md#31896) ## Related Guides * [Function Visualization](https://reference.wolfram.com/language/guide/FunctionVisualization.en.md) * [Angles and Polar Coordinates](https://reference.wolfram.com/language/guide/AnglesAndPolarCoordinates.en.md) ## History * [Introduced in 2007 (6.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn60.en.md) \| [Updated in 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) ▪ [2019 (12.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn120.en.md) ▪ [2022 (13.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn131.en.md)