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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 side-to-side?
(iii) What happens if $a$ is equal to an irrational number? Will the curve ever repeat?