Polar Coordinates Help (page 2)

Introduction to Polar Coordinates

Cartesian coordinates do not represent the only way that points can be located on a plane or in 3D space. In this chapter we’ll look at polar, latitude/longitude, cylindrical, and spherical schemes.

Two versions of the polar coordinate plane are shown in Figs. 10-1 and 10-2. The independent variable is plotted as an angle θ relative to a reference axis pointing to the right (or “east”), and the dependent variable is plotted as a distance (called the radius ) r from the origin. A coordinate point is thus denoted in the form of an ordered pair ( θ, r ).

The Radius

In any polar plane, the radii are shown by concentric circles. The larger the circle, the greater the value of r . In Figs. 10-1 and 10-2, the circles are not labeled in units. You can do that for yourself. Imagine each concentric circle, working outward, as increasing by any number of units you want. For example, each radial division might represent one unit, or five units, or 10, or 100.

Fig. 10-1 . The polar coordinate plane. The angle θ is in degrees, and the radius r is in uniform increments.

Fig. 10-2 . Another form of the polar coordinate plane. The angle θ is in radians, and the radius r is in uniform increments. 

Direction, Negative Radii, and Non-Standard Directions

The Direction

Direction can be expressed in degrees or radians counterclockwise from a reference axis pointing to the right or “east.” In Fig. 10-1, the direction θ is in degrees. Figure 10-2 shows the same polar plane, using radians to express the direction. (The “rad” abbreviation is not used, because it is obvious from the fact that the angles are multiples of π .) Regardless of whether degrees or radians are used, the angular scale is linear. The physical angle on the graph is directly proportional to the value of θ .

Fig. 10-1 . The polar coordinate plane. The angle θ is in degrees, and the radius r is in uniform increments.

Fig. 10-2 . Another form of the polar coordinate plane. The angle θ is in radians, and the radius r is in uniform increments.

Negative Radii

In polar coordinates, it is all right to have a negative radius. If some point is specified with r < 0, we multiply r by –1 so it becomes positive, and then add or subtract 180° (π rad) to or from the direction. That’s like saying, “Go 10 kilometers east” instead of “Go minus 10 kilometers west.” Negative radii must be allowed in order to graph figures that represent functions whose ranges can attain negative values.

Non-standard Directions

It’s okay to have non-standard direction angles in polar coordinates. If the value of θ is 360° (2 π rad) or more, it represents more than one complete counterclockwise revolution from the 0° (0 rad) reference axis. If the direction angle is less than 0° (0 rad), it represents clockwise revolution instead of counterclockwise revolution. Non-standard direction angles must be allowed in order to graph figures that represent functions whose domains go outside the standard angle range.