INTRODUCTION TO POLAR GRAPHS

Polar coordinates are a fundamentally different approach to representing curves in two-dimensional space. The concept is pretty easy to grasp graphically, but if you have never used polar coordinates and want to understand them, you should probably read the section below.

The traditional Cartesian method relies on an x and a y coordinate to mark how far a point is from the axes in two perpendicular directions; polar coordinates plot the location of a point by one coordinate represented by the Greek letter theta which is simplified to `t` in Graphmatica and another called `r`. The `t` tells what direction to go in from the origin, and the `r` tells how far to go out in that direction to reach the point. The direction is measured in radians as an angle starting from the positive side of the x-axis and turning around counter-clockwise (like measuring the angle the hand on a clock has traveled starting at the 3 o'clock position and going backwards). There are 2pi radians in a complete circle, corresponding to 360 of the degrees you're familiar with. To put a polar coordinate into Cartesian terms in order to graph it, we use the equations: `x = r cos t` and `y = r sin t`.

To make a graph using polar coordinates, we let theta be the independent variable and calculate a distance to plot out from the origin as we let the angle sweep around in the positive direction. The domain for the graphing is 0 to 2pi (the first complete circle in the positive direction), but you can easily change these values using the Theta Range function in the Options menu.Theta Range option in the Settings menu. Polar graphs can be typed in the equation combobox just like normal graphs. The only difference in what you type, and the way Graphmatica detects a polar graph, is that you must use the variables `t` and `r` instead of `x` and `y`. The restrictions are still the same: you can have one and only one instance of the dependent variable `r`, although it can be located almost anywhere in the equation. You can embed the `r` in a term like `r^2` to graph functions that cannot be simplified by normal means and Graphmatica will evaluate both positive and negative roots automatically. You should watch as your graph is drawn, because often the direction it is going is almost as important as the figure it draws. (When you have a "double" equation with `r^2` in it, though, note that the positive roots are drawn first and then the negative roots are drawn: theoretically they should be drawn simultaneously but this is not practically possible.)

Please note that the x and y coordinate ranges and the range for the variable theta function completely independently; in normal Cartesian graphing, theta's value is irrelevant, and in polar graphing, theta controls the domain of the graph, but the x and y ranges still control the physical screen you see. If you want to change your view of a polar graph, you use the scale or range functions just as you would normally.

See also Specifying Domains for hints on how to specify an angular domain.

kSoft, Inc. ksoft@graphmatica.com Last updated: Sun 11 Jun 2017