POLAR COORDINATES: WHAT THEY ARE AND HOW TO …

1 POLAR coordinates : what they ARE AND HOW TO USETHEMHEMANT D. TAGARE 1. note is about POLAR coordinates . I want to explain whatthey are and how to use different coordinate systems are used in mathematics and physics and allof them share some common ideas. I think it is easier to begin by understandingwhat these common features are. So I am going to introduce four common ideas ofcoordinate systems. I will first state each idea abstractly,then illustrate it by usingthe usual x,y coordinates , and finally tell you how it appliesto POLAR The coordinate system as a most basic question is: what is acoordinate system?

2 The answer is so important that I am goingto state it in boldfont:A coordinate system is a rule for mapping pairs of numbers to pointsin the may not make much sense to you right now, but you ll see what I meanshortly below when we discuss the x,y and the POLAR coordinate systems. I do wantto emphasize two things:1. A coordinate system is not just a set of axes, it is a set of rules for mappinga pair of numbers onto a point in the Different coordinate systems correspond to different rules. The POLAR coordi-nate system has rules that are different than the rules of the x,y coordinatesystem.

3 Other coordinate systems have yet other rules. Learning a new co-ordinate system comes down to understanding its rules. Keepthis in mindas you read the rest of this The x,y are the rules for the x,y coordinate system :1. Choose a point in the plane and call it theorigin. The location of this pointis arbitrary, you can choose any point as the Draw two perpendicular lines passing through the are thex and they axis. The x-axis does not have to be horizontal, nor the y-axisvertical (although that is the commonly used convention).

4 they do haveto be perpendicular. Some possible x- and y-axes are in Lookespecially at the similarity and difference between the b and c parts of thefigures. The x-axis isnotrequired to be horizontal it is only a conventionthat the x-axis is horizontal (a convention that we will follow, but only aconvention nevertheless).3. Choose one side of the x-axis as positive. The other side ofthe x-axis is neg-ative. Now rotate the positive side of the x-axis through 90 degrees counter-clockwise. The part of the y-axis that it (it = the +vex-axis) coincides withis the positive y-axis.

5 The other part of the y-axis is negative. Figure the signs of the axes from figures Notice that once you choose Copyright Hemant D. Tagare, 2006. Do not copy or distribute this work without the author [a][b][c]Fig. coordinate signs on the x-axis the signs on the y-axis are completelydetermined bythe [a][b][c]++++++------90 deg. 90 deg. 90 deg. Fig. coordinate Having chosen an origin and the axes, here is the rule for taking a pair ofnumbers say (u, v) to a unique point in the plane (illustrated in , with the x- and y-axis in the conventional position).

6 The rule is that westart from the origin, go a distanceualong the x-axis and then a distancevparallel to the y-axis. Distances are considered to have signs, so that positiveand negative distances on the x-axis are to the left and rightof the origin, andon the y-axis towards top and bottom. The point we arrive at isthe pointassociated with the pair of numbers (u, v). We say that (u, v) ismappedtothis point, or that (u, v) are thecoordinatesof this ( coordinates (u,v))X-axisY-axisOriginuvuvFig. xy coordinate The POLAR POLAR coordinates , the numbers (u, v) are in-terpretedvery differently:2 The first numberuis taken to be adistanceand the second numbervis takento be anangle(usually in radians).

7 To be explicit about this, we will denote the pairas (r, ) instead of (u, v). The numbersrand can be positive, negative or are the rules for the POLAR coordinate system:1. Choose a point in the plane as the origin and draw the x-axis. As before,you can choose any point as the origin and the x-axis is not required to behorizontal, but is conventionally chosen to be the positiveand negative sides of the x-axis with a + and a sign as below:+-Originx-axisFig. and Draw a line through the origin that makes an angle with the +ve angle is positive in thecounter clockwisedirection and negative in theclockwisedirection.

8 Call this lineL:+-x-axisLine LAngle Fig. Line Imagine rotating the x-axis through the same angle and making it coincidewith the lineL. Mark as positive the part of the lineLthat the positivex-axis coincides with and mark as negative the part that the negative x-axiscoincides with. This is similar to what we did in the x,y coordinate system:+-x-axisLine LAngle Fig. signed distance along Line Find the point onLthat is a distancerfrom the origin. Positive and negativedistances are in those parts ofLthat we marked positive and negative above(figure ).

9 The point that you marked is the point that corresponds to (r, )in the POLAR coordinate +-x-axisLine LAngle Distance rPoint with POLAR coordinates (r, )Fig. point with POLAR coordinates (r, ).That s it. That s the rule for POLAR coordinates . The numbers (r, ) are called thepolar coordinatesof the point we are some examples of plotting points using their polarcoordinates. Please try to do the examples yourself and compare the results. Keepin mind that all angles are in radians. Be sure that you can do and understand theexamples c-d (Hint: 13 /6 = 2 + /6).

10 +-x-axisLine L(r, )=(1, /4)+-x-axisLine L(r, )=(-2, /2)(a) (r, ) = (1, /4)(b) (r, ) = ( 2, /2)+-x-axisLine L(r, )=(3/2,- /4)+-x-axisLine L(r, )=(2,13 /6)(c) (r, ) = (3/2, /4)(d) (r, ) = (2,13 /6)Fig. Some properties of POLAR are some aspects of polarcoordinates that are tricky. You should pay attention to thefollowing:1. Two different POLAR coordinates , say (r1, 1) and (r2, 2), can map to the samepoint. This can happen in the following ways:(a) It can happen ifr2=r1and 2= 1 2 nfor any non zero angle 2 ncorresponds toncomplete rotations, counter clockwise fornpositive and clockwise fornnegative.