MATHEMATICA TUTORIAL

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.

Here is the curve made by taking the

coordinate of each point to be Sin

and the

coordinate to be Sin

In[1]:=

Out[1]=

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

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

direction.

In[2]:=

Out[2]=

ParametricPlot3D

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.

Here the

and

coordinates for the quadrilaterals are given simply by

and

. The result is a surface plot of the kind that can be produced by Plot3D.

In[3]:=

Out[3]=

This shows the same surface as before, but with the

coordinates distorted by a quadratic transformation.

In[4]:=

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This produces a helicoid surface by taking the helical curve shown above, and at each section of the curve drawing a quadrilateral.

In[5]:=

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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.

This produces a cylinder. Varying the

parameter yields a circle in the

plane, while varying

moves the circles in the

direction.

In[6]:=

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This produces a torus. Varying

yields a circle, while varying

rotates the circle around the

axis to form the torus.

In[7]:=

Out[7]=

This produces a sphere.

In[8]:=

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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.