MATH 261, Spring 2001 Due Date: Name(s):
Honors Project 17: Lissajoux Curves
Part I
Narrative
A Lissajoux curve is a parametrized curve of the form
x = f(m + ), y = g(n + )
where m and n are integers, and f and g are either sine or cosine functions. For example
(t) = (sin mt, sin nt), (t) = (sin mt, cos nt), (t) = (cos mt, cos nt)
are Lissajoux curves. Some Lissajoux curves are illustrated below. Lissajoux curves can be generated on
an oscilloscope by plotting a known signal (or sine wave) in one coordinate direction, against an unknown
signal (or sin wave) in the other. Experience with Lissajoux curves, together with a little trial-and-error
modification of the known signal can help the user determine the amplitude, frequency, and phase of the
unknown signal.
Task
1. Draw the graph of the Lissajoux curve x = sin m, y = sin n for various combinations of relatively
prime integers m and n. (Two integers are relatively prime if they have no common divisor other than
1.)
2. What conclusion(s) can you draw from part (1)? (Hint: Count the number of times the curves you drew
achieve their maximum x-values, the number of times they achieve their minimum x-values, the number
of times they achieve their maximum y-values, and the number of times they achieve their minimum
y-values.)
3. Repeat parts (1) and (2) above using values for m and n that are not relatively prime.
4. Repeat parts (1), (2), and (3) incorporating a phase shift in one component; that is, for curves of the
form x = sin m, y = sin(n + ) for various combinations of integers m and n and come constant .
O
r
x
y
z
T
5. What can you say about the six Lissajoux curves illustrated above if in them (reading across the first
row and then across the second), m = 2, 3, 3, 4, 4, 4?
Part II
Narrative
Lissajoux curves can be created by projecting space curves into coordinate planes: The right circular
cylinder whose cartesian equation is x2
+ y2
= 1 can be parametrized by
x(, r) = cos m, y(, r) = sin m, z(, r) = r,
and we may think of
T(, r) = (x(, r), y(, r), z(, r))
as a transformation from the (, r)-coordinate plane to Cartesian (x, y, z)-space (see the figure below). The
image of the curve in the r-plane whose equation is r = sin n has parametric equations
x() = cos m, y() = sin m, z() = sin n,
and the projection of this curve into the yz-coordinate plane has the parametric equations y = sin m, z =
sin n.
Task
1. Illustrate the above Narrative by drawing, for some fixed values of m and n, three copies of the space
curve () = (cos m, sin m, sin n), one view in perspective, one view from above ("down the z-axis"),
and one view from "down the x-axis". (One of these views should reveal a Lissajoux curve.)
2. Discuss the geometry associated with a nonzero phase angle  in r: that is, the image under T of
r = sin(n + ).
Contributor: Dr. Kris A. Dines
X-Data Corporation, Indianapolis, IN