# Parametric Plots

"Basic Plotting" describes how to plot curves in *Mathematica* in which you give the coordinate of each point as a function of the coordinate. You can also use *Mathematica* to make *parametric* plots. In a parametric plot, you give both the and coordinates of each point as a function of a third parameter, say .

ParametricPlot[{f_{x},f_{y}},{t,t_{min},t_{max}}] | |

make a parametric plot | |

ParametricPlot[{{f_{x},f_{y}},{g_{x},g_{y}},...},{t,t_{min},t_{max}}] | |

plot several parametric curves together |

Functions for generating parametric plots.

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ParametricPlot3D[{f_{x},f_{y},f_{z}},{t,t_{min},t_{max}}] | |

make a parametric plot of a three-dimensional curve | |

ParametricPlot3D[{f_{x},f_{y},f_{z}},{t,t_{min},t_{max}},{u,u_{min},u_{max}}] | |

make a parametric plot of a three-dimensional surface | |

ParametricPlot3D[{{f_{x},f_{y},f_{z}},{g_{x},g_{y},g_{z}},...},...] | |

plot several objects together |

Three-dimensional parametric plots.

ParametricPlot3D[{f_{x}, f_{y}, f_{z}}, {t, t_{min}, t_{max}}] is the direct analog in three dimensions of ParametricPlot[{f_{x}, f_{y}}, {t, t_{min}, t_{max}}] in two dimensions. In both cases, *Mathematica* effectively generates a sequence of points by varying the parameter t, then forms a curve by joining these points. With ParametricPlot, the curve is in two dimensions; with ParametricPlot3D, it is in three dimensions.

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ParametricPlot3D[{f_{x}, f_{y}, f_{z}}, {t, t_{min}, t_{max}}, {u, u_{min}, u_{max}}] creates a surface, rather than a curve. The surface is formed from a collection of quadrilaterals. The corners of the quadrilaterals have coordinates corresponding to the values of the when t and u take on values in a regular grid.

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In general, it is possible to construct many complicated surfaces using ParametricPlot3D. In each case, you can think of the surfaces as being formed by "distorting" or "rolling up" the - coordinate grid in a certain way.

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You should realize that when you draw surfaces with ParametricPlot3D, the exact choice of parametrization is often crucial. You should be careful, for example, to avoid parametrizations in which all or part of your surface is covered more than once. Such multiple coverings often lead to discontinuities in the mesh drawn on the surface, and may make ParametricPlot3D take much longer to render the surface.