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BUILT-IN MATHEMATICA SYMBOL
ParametricPlot3D[{fx, fy, fz}, {u, umin, umax}]
produces a three-dimensional space curve parametrized by a variable u which runs from 
to 
.
ParametricPlot3D[{fx, fy, fz}, {u, umin, umax}, {v, vmin, vmax}]
produces a three-dimensional surface parametrized by u and v.
ParametricPlot3D[{{fx, fy, fz}, {gx, gy, gz}...}...]
plots several objects together.
Details and Options
- No curve or surface is drawn in any regions where the corresponding
or 
evaluate to None, or anything other than real numbers. - ParametricPlot3D treats the variables u and v as local, effectively using Block.
- ParametricPlot3D has attribute HoldAll, and evaluates the
, 
, ... only after assigning specific numerical values to variables. - In some cases it may be more efficient to use Evaluate to evaluate the
, 
, ... symbolically before specific numerical values are assigned to variables. - ParametricPlot3D has the same options as Graphics3D, with the following additions and changes:
-
Axes True whether to draw axes BoundaryStyle None how to draw boundary lines for surfaces ColorFunction Automatic how to determine the color of curves and surfaces ColorFunctionScaling True whether to scale arguments to ColorFunction EvaluationMonitor None expression to evaluate at every function evaluation Exclusions Automatic u points or 
curves to excludeExclusionsStyle None what to draw at excluded points or curves MaxRecursion Automatic the maximum number of recursive subdivisions allowed Mesh Automatic how many mesh divisions in each direction to draw MeshFunctions Automatic how to determine the placement of mesh divisions MeshShading None how to shade regions between mesh divisions MeshStyle Automatic the style for mesh divisions Method Automatic the method to use for refining surfaces NormalsFunction Automatic how to determine effective surface normals PerformanceGoal $PerformanceGoal aspects of performance to try to optimize PlotLegends None legends for surfaces PlotPoints Automatic the initial number of sample points in each parameter PlotRange Automatic range of values to include PlotStyle Automatic graphics directives for the style for each object RegionFunction (True&) how to determine whether a point should be included TextureCoordinateFunction Automatic how to determine texture coordinates TextureCoordinateScaling True whether to scale arguments to TextureCoordinateFunction WorkingPrecision MachinePrecision the precision used in internal computations - Interactive labeling can be specified for curves or surfaces using Tooltip, StatusArea, or Annotation.
- ParametricPlot3D[Tooltip[list], ...] specifies that
should be displayed as tooltip labels for the corresponding curves or surfaces. - Tooltip[{fx, fy, fz}, label] specifies an explicit tooltip label for a curve or surface.
- All the functions
etc. should give real numbers for all values of parameters at which they are evaluated. There will be holes in the final surface anywhere at which 
etc. do not yield real number values. - The default setting PlotPoints->Automatic corresponds to PlotPoints->75 for curves and PlotPoints->{15, 15} for surfaces.
- ParametricPlot3D 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 in each parameter at most MaxRecursion times.
- You should realize that with the finite number of sample points used, it is possible for ParametricPlot3D to miss features in your functions. To check your results, you should try increasing the settings for PlotPoints and MaxRecursion.
- On[ParametricPlot3D::accbend] makes ParametricPlot3D 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
for surfaces. - The default setting MeshFunctions->Automatic corresponds to
for curves, and 
for surfaces. - The arguments supplied to functions in MeshFunctions and RegionFunction are x, y, z, u, and v. Functions in ColorFunction and TextureCoordinateFunction are by default supplied with scaled versions of these arguments.
- The functions are evaluated all along each curve, or all over each surface.
- By default, surfaces are treated as uniform white diffuse reflectors, corresponding to ColorFunction->(White&).
- ParametricPlot3D returns Graphics3D[GraphicsComplex[data]].
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