This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# PolarPlot

 PolarPlot generates a polar plot of a curve with radius r as a function of angle . PolarPlotmakes a polar plot of curves with radius functions , , ....
• The angle is measured in radians, counterclockwise from the positive axis.
• The , position corresponding to , is , . The value of need not be between and .
• PolarPlot has attribute HoldAll, and evaluates functions only after assigning specific numerical values to .
• In some cases it may be more efficient to use Evaluate to evaluate functions symbolically before specific numerical values are assigned to .
• No curve is drawn in any region where a function evaluates to None.
• 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 to exclude ExclusionsStyle None what to draw at excluded points 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 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 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 WorkingPrecision MachinePrecision the precision used in internal computations
• PolarPlot[Tooltip[{f1, f2, ...}], {, min, max}] specifies that the should be displayed as tooltip labels for the corresponding curves.
• Tooltip specifies an explicit tooltip label for a curve.
• 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 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 functions are evaluated all along each curve.
Make a polar plot:
Make several polar plots:
Style the curves:
Make a polar plot:
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Make several polar plots:
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Style the curves:
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 Scope   (13)
More points are sampled when the function changes quickly:
The plot range is selected automatically:
Ranges where the function becomes nonreal are excluded:
The curve is split when there are discontinuities in the function:
Use PlotPoints and MaxRecursion to control adaptive sampling:
Use PlotRange to focus in on areas of interest:
Multiple curves are automatically colored to be distinct:
Provide explicit styling to different curves and regions:
Provide an interactive Tooltip for each curve:
Create an overlay mesh:
Style the areas between mesh levels:
Color by parameter values:
 Options   (43)
Color the curve by scaled , , , or value:
ColorFunction has higher priority than PlotStyle:
Use red for the parameter :
Color the curve by angle:
Find the list of parameter values evaluated:
Find the coordinate values:
Count how many times the function is evaluated:
Automatically determine exclusions:
Provide an explicit list of points for exclusions:
Specify exclusions using equations:
Specify explicit styling for lines joining exclusion points:
Provide styling for both exclusion points and the lines joining them:
Each level of MaxRecursion will adaptively subdivide the initial mesh into a finer mesh:
Show the initial and final sampling meshes:
Use 10 mesh points evenly spaced in the direction:
Use an explicit list of values for the mesh in the direction:
Use explicit value and style for the mesh:
Use a mesh evenly spaced in the , , , and directions:
Show five mesh levels in the direction (red) and ten in the direction (blue):
Alternate red and blue arcs in the direction:
Use None to remove segments:
MeshShading can be used with PlotStyle:
MeshShading has higher priority than PlotStyle for styling:
Use PlotStyle for some segments by setting MeshShading to Automatic:
MeshShading can be used with ColorFunction:
Automatically choose the mesh style:
Use a red mesh in the direction:
Use a red mesh in the direction and a blue mesh in the direction:
Use big red mesh levels in the direction:
Generate a higher-quality plot:
Emphasize performance, possibly at the cost of quality:
Use more initial points to get a smoother plot:
Show the curve where and :
With the natural range of values, the fine detail around the origin is not visible:
Use PlotRange to focus in on areas of interest:
Use different style directives:
By default different styles are chosen for multiple curves and regions:
Explicitly specify the style for different curves and regions:
PlotStyle can be combined with ColorFunction:
PlotStyle can be combined with MeshShading:
Show the plot where :
Evaluate functions using machine-precision arithmetic:
Evaluate functions using arbitrary-precision arithmetic:
 Applications   (4)
Plot a circle:
A spiral:
An oscillation around a circle:
Archimedean spirals of the form :
Archimedean spirals of the form :
Logarithmic spirals have the form :
Create a Guilloché pattern :
PolarPlot is a special case of ParametricPlot for curves:
Use ListPolarPlot for data:
Use Plot3D and ParametricPlot3D for function and parametric surfaces:
Use RevolutionPlot3D and SphericalPlot3D for cylindrical and spherical coordinates:
Use ContourPlot and RegionPlot for implicit curves and regions: