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### Coordinate Systems

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Date: 04/22/97 at 17:39:41
From: Chris Eagle
Subject: Coordinate systems

What is the polar coordinate system and how does it differ from the
rectangular coordinate system?
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```
Date: 05/22/97 at 22:31:51
From: Doctor Sydney
Subject: Re: Coordinate systems

Dear Chris,

Hello!  This is a good question.  When you first come across the polar
coordinate system, it might seem a bit strange, but once you are more
familiar with it, you will probably begin to understand it better, and
like it!

The polar coordinate system and the rectangular coordinate system are
both designed to label points in the plane.  They differ in the ways
they represent points in the plane.

Let's first think about the rectangular coordinate system.  We define
a point (x,y) in the rectangular coordinate system to be the point we
get to when we move from the origin horizontally x units and
vertically y units, right?  For instance, the point (1,-2) is the
point in the fourth quadrant that is 1 unit to the right and 2 units
below the origin. So, each of the rectangular coordinates of a point
gives you information about where that point is located.

In a similar way, the polar coordinates of a point gives you
information about where that point is located. However, polar
coordinates work under a different system than rectangular
coordinates. Suppose we were given a point (x,y) in polar coordinates.
How would we find this point on a graph?  Well, the way that polar
coordinates are defined, we would start at the origin, find the ray
the emanates from the origin that is an angle of y with the positive
x-axis, and then go out a distance of y on this ray. For instance, if
we wanted to find the point (1, pi/2) where pi/2 is the angle that
corresponds to 90 degrees, we would go out a distance of 1 on the ray
that is 90 degrees from what would be the positive if we were working
in rectangular coordinates.

Because these systems are so different, there are different types of
graph paper for them. When we are graphing things using the
rectangular coordinate system, we use "standard" grid graph paper.
This makes sense since when using the rectangular coordinate system,
all we ever do to find points is move horizontally and vertically.
However, when we graph  using the polar coordinate system, the graph
paper usually has circles whose center is the origin, and rays that
emanate from the origin.  This makes it easier to figure out where
points in polar coordinates lie. Do you see why?

There are lots of fun things you can do with polar coordinates. For
between two points in polar coordinates? In other words, if you are
given two points, (a,b) and (c,d), in polar coordinates, what is the
distance between the two points?  It isn't sqrt[(d-b)^2 + (c - a)^2].
What is it?  Draw a graph to figure it out!

Also, the existence of multiple ways to assign pairs of numbers to
points in the plane raises some interesting questions. Think about
these questions for fun: Both the rectangular coordinate system and
the polar coordinate system assign PAIRS of numbers to points in
space. Could we define a coordinate system that assigns only SINGLE
numbers (instead of pairs of numbers) to points in the plane?  If so,
would this coordinate system cover the whole plane or would it cover
only part of the plane? Could we define a coordinate systsem that
assigns TRIPLES to points in the plane?  What if we moved from the
plane to three-dimensional space. How many coordinates are necessary
in a coordinate system in three-dimensional space?

If you want more problems to think about, if you have questions about
some of the questions I mentioned at the end of the message, or if you
need more help understanding the difference between polar and
rectangular coordinates, please do write back.  Good luck!

-Doctor Sydney,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Coordinate Plane Geometry
High School Definitions
High School Geometry

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