Assignment 10
Parametric Curves


by: Kelli Nipper

A parametric curve is a pair of functions x=f(t), y=g(t) that give x and y as contiuous functions of the real number t (the parameter). Each value determines a point ( f(t), g(t) ). The graph is the set of all such points in the interval assigned.

x= cos t

y= sin t

[0, 2 pi]


The point (cos t, sin t) begins at (1, 0). As t moves counterclockwise from 0 to 2 pi, it gives points that satisfy: x = cos t, y = sin t

All points lie on the unit circle, thus the graph is the unit circle.

A given figure in the plane may have different parameterizations. For example:


This parametric curve also lies on the unit circle because

To explore other graphs I changed the parameters:

For various a and b, x = cos (at) ; y = sin (bt)

 

Changes the period.

For various a and b, x = a cos (t) ; y = b sin (t)

Changes the amplitude. In general, when a and b increase, the amplitude increases. When a and b decreases, the period decreases.

In closing, to describe arbitrary amplituted and periods, for x = a sin bt and y = a cos bt

a is the amplitude and 2pi/b is the period.

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