Built-in Mathematica Symbol

ParametricPlot

ParametricPlot[{f_x, f_y}, {u, u_min, u_max}]
generates a parametric plot of a curve with x and y coordinates f_x and f_y as a function of u.

ParametricPlot[{{f_x, f_y}, {g_x, g_y}, ...}, {u, u_min, u_max}]
plots several parametric curves.

ParametricPlot[{f_x, f_y}, {u, u_min, u_max}, {v, v_min, v_max}]
plots a parametric region.

ParametricPlot[{{f_x, f_y}, {g_x, g_y}, ...}, {u, u_min, u_max}, {v, v_min, v_max}]
plots several parametric regions.

ParametricPlot has attribute HoldAll, and evaluates the f_i and g_i only after assigning specific numerical values to variables.

In some cases it may be more efficient to use Evaluate to evaluate the f_i and g_i symbolically before specific numerical values are assigned to variables.

No curve is drawn in any regions where the corresponding f_i or g_i evaluate to None.

ParametricPlot has the same options as Graphics, with the following additions and changes:

AspectRatio	Automatic	ratio of height to width
Axes	True	whether to draw axes
BoundaryStyle	Automatic	how to draw boundaries of regions
ColorFunction	Automatic	how to apply coloring to curves or regions
ColorFunctionScaling	True	whether to scale arguments to ColorFunction
EvaluationMonitor	None	expression to evaluate at every function evaluation
Exclusions	Automatic	u points or u, v curves to exclude
ExclusionsStyle	None	what to draw at excluded points or curves
Frame	Automatic	whether to put a frame around the plot
MaxRecursion	Automatic	the maximum number of recursive subdivisions allowed
Mesh	Automatic	how many mesh divisions to draw
MeshFunctions	Automatic	how to determine the placement of mesh divisions
MeshShading	None	how to shade regions between mesh points or lines
MeshStyle	Automatic	the style for mesh divisions
Method	Automatic	the method to use for refining curves or regions
PerformanceGoal	$PerformanceGoal	aspects of performance to try to optimize
PlotPoints	Automatic	initial number of sample points in each parameter
PlotRange	Automatic	range of values to include
PlotRangeClipping	True	whether to clip at the plot range
PlotStyle	Automatic	graphics directives to specify the style for each object
RegionFunction	(True&)	how to determine whether a point should be included
WorkingPrecision	MachinePrecision	the precision used in internal computations

Interactive labeling can be specified for curves and regions using Tooltip, StatusArea or Annotation.

ParametricPlot[Tooltip[{{f_x, f_y}, ...}, ...]] specifies that {f_x, f_y} should be displayed as tooltip labels for the corresponding curves or regions.

Tooltip[{f_x, f_y}, label] specifies an explicit tooltip label for a curve or region.

ParametricPlot initially evaluates each function 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 in each parameter at most MaxRecursion times.

You should realize that with the finite number of sample points used, it is possible for ParametricPlot to miss features in your functions. To check your results, you should try increasing the settings for PlotPoints and MaxRecursion.

On[ParametricPlot::accbend] makes ParametricPlot print a message if it is unable to reach a certain smoothness of curve.

The default setting Mesh->Automatic corresponds to None for curves, and 15 for regions.

With Mesh->All, ParametricPlot will explicitly draw a point at each sample point on each curve, or will draw a line to indicate each region subdivision.

The default setting MeshFunctions->Automatic corresponds to {#3&} for curves, and {#3&, #4&} for regions.

The arguments supplied to functions in MeshFunctions and RegionFunction are x, y, u and v. Functions in ColorFunction are by default supplied with scaled versions of these arguments.