Talk:Polar coordinate system/draft

Applications
Polar coordinates are two-dimensional and thus their use can only be in situations where point positions lie on a single two-dimensional plane. They are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point. For instance, the examples above show how elementary polar equations suffice to define curves such as the Archimedean spiral whose equation in the Cartesian coordinate system would be much more intricate. Moreover, many physical systems such as those concerned with bodies moving around a central point or with phenomena originating from a central point are simpler and more intuitive to model using polar coordinates. The initial motivations for the introduction of the polar system was the study of circular and orbital motion.

Position and navigation
Polar coordinates can be used whenever real-world positions are given by angle and distance from a central point. Polar coordinates are used often in navigation, as the destination or direction of travel can be given as an angle and distance from the object being considered. Aircraft use a slightly modified version of the polar coordinate system for navigation. In this coordinate system, the 0° ray (generally called heading 360) is vertical, and the angles continue in a clockwise, rather than a counterclockwise, direction, as is usual in navigational coordinate systems. Heading 360 corresponds to magnetic north, while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively. Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read niner-zero by air traffic control).

Many robots capable of movement also use the polar coordinate system (or a slightly modified version of it) for navigation. This is very convenient for artificial intelligence, as the center of the coordinate system (the pole) can always be placed at the robot's present location, with only the distance and direction of travel left for the robot to calculate. Robots with a stationary rotating base and extending limbs can also use the polar coordinate system to calculate how far and in which direction to extend the limb.

Modeling
Systems displaying radial symmetry provide natural settings for the polar coordinate system, with the central point acting as the pole. A prime example of this usage is the Groundwater flow equation when applied to radially symmetric wells.

Systems with a radial force are also good candidates for the polar coordinate system. These systems include gravitational fields, which obey the inverse-square law, as well as systems with point sources of any kind.

Even radially asymmetric systems may be easily modeled with polar coordinates, however. For example, microphone polar patterns illustrate the locus of points that produce the same signal level output in the microphone if a given sound pressure level is generated from that point. If the microphone represents the origin in a polar coordinate plane, and the forward direction is considered to be the 90° ray, these loci can be represented as polar curves. For example, this locus for the cardioid microphone, the most common unidirectional microphone, can be represented as r = a + a sin θ, with the value of a depending on the sound pressure level.