Plotting Parametric Curves with MATHEMATICA

On this page we describe the MATHEMATICA commands ParametricPlot and ParametricPlot3D which can be used for plotting parametrically defined curves in 2- and 3-dimensions. We start with the command for sketching planar curves:

This will draw the curve described by the parametric equations { x = x(t), y = y(t) } as t ranges through the interval [tmin,tmax] . As with the Plot command, two or more curves can be plotted with

For example the output of

is:

To get a fancier output we might add some modifiers such as

obtaining:

Some of the most useful modifiers for the ParametricPlot command are:

In this table, NN denotes a numerical value (which should be between 0 and 1 for Hue[NN]). The symbol BB can be one of the values True or False. And s₁ may include specifications such as Hue[NN] (setting curve color), AbsoluteThickness[NN] (setting curve thickness), or Dashing[{NN,NN}] (making the curve dashed). The modifier AspectRatio->Automatic gives the visually true proportions (where the x- and y-axes are scaled equally). Instead of Hue[NN], colors can also be specified with RGBColor[NN,NN,NN]. Desired colors may be previewed and chosen using the "Color Selector" from the "Input" menu.

MATHEMATICA can also animate graphics to represent the motion of a parametrization. For example:

  Table[
    ParametricPlot[{Sin[2t], Cos[5t]}, {t, 0, I}, 
      PlotStyle -> {Hue[.77], Thickness[.01]}, 
      PlotRange -> {{-1.2, 1.2}, {-1.2, 1.2}}, Frame -> True, 
      Background -> Hue[.15]],
  {I, 0, 2*Pi, 2*Pi/20}]

will result in the animation

Or check out this more intricate example plotting two curves simultaneously.

The basic command for sketching a parametrized 3-dimensional curve is:

which draws the curve described by the equations { x = x(t), y = y(t), z = z(t) } as t ranges through the interval [tmin,tmax] .

Here's an illustration:

  ParametricPlot3D[ 
    {Sin[2t], Cos[5t],t,{Hue[.75], Thickness[.01]}}, {t,0,2*Pi}, 
    BoxRatios->{1,1,1},ViewPoint->{2, 3, 0},
    Background->Hue[.07]   ]

which results in

The modifiers for ParametricPlot3D are similar to those for ParametricPlot with a few variations:

3-dimensional pictures can be hard to interpret at times, it may help to spin the picture in order to gain perspective. Here's a spinning example to illustrate.