Friday, December 16, 2011

Graphing - Polar Coordinates

Polar coordinates are a system of describing points on a plane, in a way that is similar yet still quite different from what you have previously used.  I am going to explain what polar coordinates are here, and then in a subsequent post, I am going to explain their relationship to the graphing system that you already know about.  Hopefully you will find that graphing polar coordinates are not really any harder than what you already know how to do!

Up until now, you have likely only ever worked with rectangular coordinates, otherwise known as the Cartesian coordinate system.  This is the familiar horizontal x-axis and vertical y-axis, which intersect at the origin and separate the coordinate plane into four quadrants.  Any point on the xy plane can be described by it's location relative to these axes, which requires knowing how far horizontal and how far vertical it is away from the origin.  These are the x-coordinate and y-coordinate of the point, and together they form an ordered pair, which completely describes the point's location on the xy plane.  (For those looking for extra trivia: the x-coordinate has the technical name "abscissa" while the y-coordinate is called the "ordinate."  Try explaining that to your teacher for bonus points!  Also of interest, this Cartesian coordinate system is named after René Descartes, who was a French philosopher and mathematician whose work, among other things, first described the connection between algebra and geometry.)

Polar form is quite different from this rectangular coordinate system.  It requires a different way of thinking about the points on the graph.  Don't think of an x-y axis system for now.  To start, just place a point, and then from that point extend a line out to the right (this is conventionally what is done, though technically you could draw it in any direction).  The starting point, O, is the origin of this coordinate system, but in this case it may also rightfully be called the pole.  The line that you have extended from the pole is the polar axis.  We will define points on the plane relative to this axis (as opposed to the Cartesian system, which defines points relative to two axes).

So then, now we need a point P to talk about.  Pick any point, and then draw a line (called r, as in radius) that connects it to the pole.  The basic premise of a polar coordinate graph is that you describe the point by providing its distance from the pole, and also the angle (ɵ, theta) by which r is rotated away from the polar axis.  Knowing these two dimensions is enough information to fully describe where your point is located on the plane.  So, while a Cartesian ordered pair looks like (x, y), a polar ordered pair looks like (r, ɵ).  So, r and ɵ are referred to as polar coordinates of P, and it is written as P(r, ɵ) which explains that P is the point with polar coordinates (r, ɵ).

To assist you in drawing your polar graphs, polar coordinate graph paper is available.  Instead of being labeled as x and y axes, they are labeled with angles (in radians... I'll cover radians in another post as well).  However, I don't think you really need to use it until you start graphing more in-depth polar equations.  Whenever I draw a polar graph, I just draw my own.  :)  Briefly, 180°= π radians, 90° = π/2 radians, 270° = 3π/2 radians, and 360° = 2π radians.  I'm not going to go into anymore detail than this about radians, because if you're at the point where you're studying polar coordinate graphs, you've already covered radians.  :)

Using this new coordinate system and working with polar equations will allow you to draw some really cool polar graphs.  Expect to see plenty of "four-leaf clover" graphs and "spiral" graphs, among other really interesting designs.  Just try to think of your graph as a series of points along a line that is rotating or sweeping around the pole in the center, moving closer or further as it revolves around it, as described by its equation.  I will explain some polar equations in a future post that will highlight some common and cool polar graphs.

In my next post, I want to explain to you how to convert from polar coordinates to rectangular coordinates.  With a little bit of logic, you will see that the derivation to connect polar form to rectangular form is quite easy.  Similarly, going from rectangular coordinates to polar coordinates follows along the same theory.  This connection is a helpful basis to build upon, since by this point you are already very familiar with the Cartesian coordinate system.  Layering the polar coordinate system on top is the best way to learn this new approach to graphing.

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