**The Polar Plane**

The *polar coordinate plane* is an alternative way of expressing the positions of points and of graphing equations and relations in two dimensions. The independent variable is plotted as the distance or radius *r* from the origin, and the dependent variable is plotted as an angle *q* relative to a reference axis. Figure 3-11 shows the polar plane that is used most often by physicists. The angle *q* is expressed in units called *radians* . One radian is the angle defined by a circular arc whose length is the same as the radius of the circle containing that arc. If this is too complicated to remember, then think of it like this: A radian is a little more than 57°. Or you can remember that there are 2π, or about 6.28, radians in a complete circle. The angle *q* is plotted counterclockwise from the ray extending to the right.

Figure 3-12 shows the polar system employed by some engineers, especially those in communications. Navigators and astronomers often use this scheme too. The angle *q* is expressed in degrees here and is plotted clockwise from the ray extending upward (corresponding to geographic north). You have seen this coordinate system in radar pictures of storms. If you’re in the military, especially in the Navy or the Air Force, you’ll know it as a polar radar display. Sometimes this type of polar display shows the angle referenced clockwise from south rather than from geographic north.

**Equation Of Circle Centered At The Origin**

The equation of a *circle* centered at the origin in the polar plane is just about as simple as an equation can get. It is given by the following formula:

*r* = *a*

where *a* is a real number and *a* > 0. This is shown in Fig. 3-13. Certain other graphs, such as cloverleaf patterns, spirals, and cardioids (heart-shaped patterns), also have simple equations in polar coordinates but complicated equations in rectangular coordinates.

Practice problems for these concepts can be found at: Graphing Schemes for Physics Practice Test

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