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ParametricPlot3D[{fx, fy, fz}, {u, umin, umax}]
produces a three-dimensional space curve parametrized by a variable u which runs from umin to umax.
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.
  • No curve or surface is drawn in any regions where the corresponding fi or gi evaluate to None, or anything other than real numbers.
  • ParametricPlot3D has attribute HoldAll, and evaluates the f_i, 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, g_i, … symbolically before specific numerical values are assigned to variables.
AxesTruewhether to draw axes
BoundaryStyleNonehow to draw boundary lines for surfaces
ColorFunctionAutomatichow to determine the color of curves and surfaces
ColorFunctionScalingTruewhether to scale arguments to ColorFunction
EvaluationMonitorNoneexpression to evaluate at every function evaluation
ExclusionsAutomaticu points or u, v curves to exclude
ExclusionsStyleNonewhat to draw at excluded points or curves
MaxRecursionAutomaticthe maximum number of recursive subdivisions allowed
MeshAutomatichow many mesh divisions in each direction to draw
MeshFunctionsAutomatichow to determine the placement of mesh divisions
MeshShadingNonehow to shade regions between mesh divisions
MeshStyleAutomaticthe style for mesh divisions
MethodAutomaticthe method to use for refining surfaces
NormalsFunctionAutomatichow to determine effective surface normals
PerformanceGoal$PerformanceGoalaspects of performance to try to optimize
PlotPointsAutomaticthe initial number of sample points in each parameter
PlotRangeAutomaticrange of values to include
PlotStyleAutomaticgraphics directives for the style for each object
RegionFunction(True&)how to determine whether a point should be included
WorkingPrecisionMachinePrecisionthe precision used in internal computations
  • ParametricPlot3D[Tooltip[list], ...] specifies that {f_x,f_y,f_z} 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 f_x 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 f_x etc. do not yield real number values.
  • 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 15 for surfaces.
  • 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&).
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