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A repost from alt.uu.future,alt.education.distance:

From: esz001@cck.coventry.ac.uk (Will Overington)
Message-ID: <4y1mb57d@cck.coventry.ac.uk>
Date: 20 Jul 92 13:42:12 GMT
Organization: Coventry University, Coventry, UK


LEARNING MATERIAL UNDER DEVELOPMENT

CHEBYSHEV POLYNOMIALS

UNIT 2

20th July 1992

The purpose of this unit is to invite you to explore the
connection between Chebyshev Polynomials and Lissajous figures.

A Lissajous figure may be produced by taking a variable t, which
is time in the production of a Lissajous figure on the screen of
an oscilloscope as an electronics experiment, and which may as
well be thought of as time in a pencil and paper experiment,
though it may also be thought of as a parametric parameter if you
wish, but it is no great matter to our pencil and paper
experiment.  A point (x,y) is then plotted for each value of t,
starting say with t=0 and gradually increasing t.

The value of x is given by

x = sin(a*t)

and the value of y is given by

y = sin(b*t + p)

If one chooses a and b to be numbers such that a/b or b/a is some
fraction with small integers in it, such as 2/3 or 1/4, then
diagrams where the locus travels over and over the same points as
t increases towards infinity are achieved.

As an exercise you might like to try drawing out the Lissajous
figure where

x = sin(t)

and

y = sin(2 * t  +  20)

taking t to be in degrees, starting at 0 degrees and increasing
by 10 degrees at each step, and taking the 20 here stated as 20
degrees.  Keep on going until the curve is clearly going over the
same path again.

A second Lissajous figure that you might like to try is where

x = sin(2 * t)

and

y = sin(3 * t  +  40)

Draw this Lissajous figure on a separate sheet of graph paper.

Try several of your own devising, observing how there are
usually, but not always, loops, and how sometimes the loops are
symmetrical but that often they are not symmetrical.

Now, continue drawing Lissajous figures but use cosines instead
of sines.  Now, sines and cosines are interchangeable with an
alteration of a phase shift.  However, continue to use the cosine
form of the periodic function.

Now start to concentrate on drawing Lissajous figures of the form
that follows.

The value of x is given by

x = cos(t)

and the value of y is given by

y = cos(n * t)

where n is an integer.

Try drawing such Lissajous figures where n=0, n=1, n=2, n=3 and
n=4.

Compare these Lissajous figures with the graphs of Chebyshev
Polynomials that you drew in Unit 1 of this course.

As a final exercise, try to eliminate t from the two equations
above (that is, the equations involving cos) to produce an
equation for the nth Chebyshev Polynomial Tn(x) as a function of
x.

A set of tables of theta, with theta in degrees, together with
values of the sine and cosine of theta is included.

     theta     sine     cosine

    0.0000    0.0000    1.0000
   10.0000    0.1736    0.9848
   20.0000    0.3420    0.9397
   30.0000    0.5000    0.8660
   40.0000    0.6428    0.7660
   50.0000    0.7660    0.6428
   60.0000    0.8660    0.5000
   70.0000    0.9397    0.3420
   80.0000    0.9848    0.1736
   90.0000    1.0000   -0.0000
  100.0000    0.9848   -0.1736
  110.0000    0.9397   -0.3420
  120.0000    0.8660   -0.5000
  130.0000    0.7660   -0.6428
  140.0000    0.6428   -0.7660
  150.0000    0.5000   -0.8660
  160.0000    0.3420   -0.9397
  170.0000    0.1736   -0.9848
  180.0000   -0.0000   -1.0000
  190.0000   -0.1736   -0.9848
  200.0000   -0.3420   -0.9397
  210.0000   -0.5000   -0.8660
  220.0000   -0.6428   -0.7660
  230.0000   -0.7660   -0.6428
  240.0000   -0.8660   -0.5000
  250.0000   -0.9397   -0.3420
  260.0000   -0.9848   -0.1736
  270.0000   -1.0000    0.0000
  280.0000   -0.9848    0.1736
  290.0000   -0.9397    0.3420
  300.0000   -0.8660    0.5000
  310.0000   -0.7660    0.6428
  320.0000   -0.6428    0.7660
  330.0000   -0.5000    0.8660
  340.0000   -0.3420    0.9397
  350.0000   -0.1736    0.9848
  360.0000    0.0000    1.0000