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
coordinates of each point as a function of a third parameter, say
|make a parametric plot|
|plot several parametric curves together|
Functions for generating parametric plots.
Here is the curve made by taking the
coordinate of each point to be Sin
coordinate to be Sin
|make a parametric plot of a three-dimensional curve|
|make a parametric plot of a three-dimensional surface|
|plot several objects together|
Three-dimensional parametric plots.
is the direct analog in three dimensions of ParametricPlot
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.
This makes a parametric plot of a helical curve. Varying
produces circular motion in the
plane, and linear motion in the
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
take on values in a regular grid.
coordinates for the quadrilaterals are given simply by
. The result is a surface plot of the kind that can be produced by Plot3D
This shows the same surface as before, but with the
coordinates distorted by a quadratic transformation.
This produces a helicoid surface by taking the helical curve shown above, and at each section of the curve drawing a quadrilateral.
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.
This produces a cylinder. Varying the
parameter yields a circle in the
plane, while varying
moves the circles in the
This produces a torus. Varying
yields a circle, while varying
rotates the circle around the
axis to form the torus.
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.