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### Understanding Parametric Equations

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Date: 04/02/98 at 20:29:40
From: Kristen Demergian
Subject: Parametric Equations

Dear Dr. Math,

I'm a junior at Granite Hills High School In El Cajon. For honors
Pre-Calculus we are required to write an essay on an advanced
calculus concept that relates to the real world. I have obtained
information about parametric equations that relate them to scientific
technology currently being used today. I have been researching the
topic for about a week and am getting ready to write an outline. I
understand that parametric equations are, in general, graphs and
equations that use three variables, x, y, and t. However, I am
confused on their basic principles. Everything I've come across has
been written in advanced terminology, hence I am a little confused. I
would appreciate it if you would explain, in general, what parametric
equations are.
```

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Date: 04/03/98 at 08:08:59
From: Doctor Jerry
Subject: Re: Parametric Equations

Hi Kristen,

Only relatively few curves in the plane can be described as the graph
of an equation y = f(x). A circle can't be described in this way, just
parts of it. Parametric equations can be used to describe circles,
and much more.

Parametric equations are like this: one gives the x and y coordinates
of points on the curve in separate equations. For example, the
parametric equations:

x = cos(t)
y = sin(t)

describe a circle, as t varies over [0,2pi]. For each value of t,
plot the point (x,y) = (cos(t),sin(t)). The moving point will trace a
circle. Notice, for example, that x^2+y^2 = cos^2(t)+sin^2(t) = 1.
Hence, (x,y) is on a circle with radius 1, centered at the origin.

Another example:

x = t*cos(t)
y = t*sin(t)

is a spiral, and:

x = t-cos(t)
y = 1-sin(t)

is a cycloid curve.

Parametric curves are used in three dimensions as well. The
parametric equations:

x = cos(t)
y = sin(t)
z = t

describe a helix, curling around the z-axis.

The variable t is a parameter, often describing time. The position of
a particle at any time t might be:

x = 4t-3
y = -t^2+7t+1

which is a parabolic path. Perhaps I've said enough for you to get the
idea.

-Doctor Jerry,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Equations, Graphs, Translations

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