a parametric curve?
A parametric curve in the plane
is a pair of functions, such as:
where the two continuous functions
define ordered pairs (x,y). The two functions (or equations)
are usually called the parametric equations of a curve.
a parametric curve look like geometrically?
Well, each distinct pair of functions
(or equations) will produce a unique geometric interpretation.
We will use Graphing Calculator 3.2 software to investigate the
following parametric curve.
In this software, equations for
parametric curves must be entered in vector form. Let's investigate
the following two equations:
when these equations or their respective angles are magnified?
Let's consider the following general
examples when a = b, respectively:
Example 1 (when a and b are
multiplied with the functions, respectively)
There is a lot going on here that
one may not realize upon first glance. For example, where a =
b and a, b are equal to or greater than zero, the parametric curve
will simply expand its "radius" equal to a, b. If a
= 1, b = 1, then the parametric curve will be a circle centered
at (0,0) with radius = 1. If a = 2, b = 2, then the parametric
curve will be a circle centered at (0,0) with radius = 2, as observed
above. And so on...
What would happen if a did
not equal b? Let's hold a = 1 and vary b from 1 to 3:
Now, hold b = 1 and vary a from
1 to 3:
Now, let's consider the parametric
curves when a, b magnify the respective angles of each function.
Example 2 (when a and b are
multiplied with the angles of the functions, respectively)
When a = b, the graph will always
look like the one above! On the contrary, when a is held constant
and b varies from 1 to 10, the parametric curve changes in this way. Likewise, when b is held constant
and a varies from 1 to 10, the parametric curve changes in this way.
All of the above observations
are general manipulations of the parametric curve created by the
cos (t) and sin (t) functions.
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