Coordinate SystemsDate: 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? 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 instance, think about this: how would you figure out the distance 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/ |
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