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Understanding Parametric Equations

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.

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 

-Doctor Jerry,  The Math Forum
Check out our web site!   
Associated Topics:
High School Equations, Graphs, Translations

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