A Parameterized curve, or the parameterization of the curve refers to the map gained from the interval of the parameter.
r(t) = hx(t), y(t)i
R = [a, b]
Here, the functions x(t), y(t), are called the coordinated functions, the concept is essentially developed for the representation of the curve in the space, where r is the paremetrized curve. Therefore, the image of parametrization is known as the parameterized curve.
Therefore, for a simple understanding, the process of parameterization refers to the process of finding equations of a curve or a surface. It is also important for the understanding and evaluating the complex processes and problems.
Parameterizing a curve is the technique using which the location of every point on a curve can be described. The position of every point using this method must be one-to-one and onto, which means no point should be described more than once.
While parameterizing a curve (in Real, two and three dimensional regions), we use one parameter. This parameter is chosen to make the explanation of the points of the curve as simple as possible.
x2+y2=16 ---- (1)
we can see that y=+(16-x2)(1/2)
So , the curve in (1) can be easily described by either
r1(x)=(x,( 16-x2)(1/2)) , -4<x<4 -----(2)
r2(x)=(x,-( 16-x2)(1/2)), -4<x<4 -----(3)
complete parametrization of (1) is given by (2) and (3). But it can be more easily explained and that is shown below,
The following picture will provide a better idea of the concept. In the above picture, the graph of the function that has been parameterized is an ellipse, and the green curve is the graph of the function,
c (t) = (3cos t, 2sin t). This particular function therefore, parameterizes an ellipse.
However, parameterization is not possible in a higher dimensional space, it can only be done in the three dimensional space, input or output.