## Thursday, June 30, 2011

### Parametric

A pair or a set of parametric equations may look like
$x =2,y=t, \; t\; \in\; (-\infty,\infty)$,
or
$x =2\cos(t),y=3\sin(t), \; t\; \in\; [0,2\pi]$,
or
$x=\theta,\; \; y=2\sin \theta,\; \; \theta\; \in\; [0,2\pi]$,
or
$x=\frac{1}{2}t,\; \; y=2t,\; \; z=t,\; t\in\; (-\infty,\infty)$.

See the common feature?

Now, let's try to combine the variables:
a) $x =2,y=t, \; t\; \in\; (-\infty,\infty)\; \; \Leftrightarrow \; \; x=2$

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

This is a vertical line x=2.

b) $x =2\cos(t),y=3\sin(t), \; t\; \in\; [0,2\pi] \Leftrightarrow \frac{x^2}{4}+\frac{y^2}{9}=1$

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This is a vertical ellipse centered at the origin with vertices at (0,3) and (0,-3), and endpoints of minor axis at (2,0) and (-2,0).

c) $x=\theta,\; \; y=2\sin \theta,\; \; \theta\; \in\; [0,2\pi]\Leftrightarrow y=2\sin x$

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

This is a sine function with magnitude 2.

d) $x=\frac{1}{2}t,\; \; y=2t,\; \; z=t,\; t\in\; (-\infty,\infty) \Leftrightarrow z^2=xy$
This is an object in 3-dimension.

You see, parametric equations can form a lot of curve with different type:
a) function, non-function;
b) line, curve;
c) 2-D, 3-D;
etc.

Can you try to build one set of parametric equations and figure out the graph of the set of equations?