M408M Learning Module Pages
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Chapter 10: Parametric Equations
and Polar Coordinates
Learning module LM 10.1:
Parametrized Curves:
3 kinds of functions, 3 kinds of curves
The cycloid
Visualizing parametrized curves
Tracing curves and ellipses
Lissajous figures
Learning module LM 10.2: Calculus with Parametrized Curves:
Learning module LM 10.3: Polar Coordinates:
Learning module LM 10.4: Areas and Lengths of Polar Curves:
Learning module LM 10.5: Conic Sections:
Learning module LM 10.6: Conic Sections in Polar Coordinates:
Chapter 12: Vectors and the Geometry of Space
Chapter 13: Vector Functions
Chapter 14: Partial Derivatives
Chapter 15: Multiple Integrals


Lissajous figures
Lissajous Figures
In tracing the circle $x=\cos(t)$, $y=\sin(t)$, both the $x$ motion
and the $y$ motion repeat every $2\pi$. This means that we trace the
same circle over and over and over again. If we change the frequency
of one of the two motions, then we get a more complicated curve called
a Lissajous figure. In this demonstration we look at curves of the
form $$x=\cos(t); \qquad y=\sin(at).$$
(i) First set $a$ equal to a small integer, like 1, 2 or 3.
These are the simplest kinds of Lissajous figures.
(ii) Now set $a$ equal to a simple fraction, like $3/2$ or $5/2$ or
$2/3$ or $3/4$. What happens? When $a=p/q$, how many times does the
curve go up and down before it repeats? How many times does it go
sidetoside?
(iii) What happens if $a$ is equal to an irrational number? Will
the curve ever repeat?


