Usefulness
This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves termwise. Thus, one can describe the velocity of a particle following such a parametrized path as:
and the acceleration as:
In general, a parametric curve is a function of one independent parameter (usually denoted t). For the corresponding concept with two (or more) independent parameters, see Parametric surface.
Another important use of parametric equations is in the field of computer aided design (CAD). For example, consider the following three representations, all of which are commonly used to describe planar curves.
| Type | Form | Example | Description |
|---|---|---|---|
| 1. Explicit | Line | ||
| 2. Implicit | Circle | ||
| 3. Parametric | ; |
|
Line Circle |
The first two types are known as analytical or nonparametric representations of curves, and, in general tend to be unsuitable for use in CAD applications. For instance, both are dependent upon the choice of coordinate system and do not lend themselves well to geometric transformations, such as rotations, translations, and scaling. In addition, the implicit representation is awkward for generating points on a curve because x values may be chosen which do not actually lie on the curve. These problems are eliminated by rewriting the equations in parametric form.
Read more about this topic: Parametric Equation
Famous quotes containing the word usefulness:
“The usefulness of madmen is famous: they demonstrate society’s logic flagrantly carried out down to its last scrimshaw scrap.”
—Cynthia Ozick (b. 1928)
“But, in history, practical usefulness never determines the moral value of an achievement. Only the person who increases the knowledge humanity has about itself and enhances its creative consciousness permanently enriches humanity.”
—Stefan Zweig (18811942)
“A public man must never forget that he loses his usefulness when he as an individual, rather than his policy, becomes the issue.”
—Richard M. Nixon (1913–1995)