Wednesday, 10 July 2013
Give only two real life examples of polar coordinates.
Answer
Polar coordinates are used often in navigation. Aircraft use a slightly modified version of the polar coordinates for ascertaining their position, planning and following a rout. In addition to that, polar coordinates can be used only where point positions lie on a single two-dimensional plane.
Answer 2:
Polar coordinates are two-dimensional and thus they can be used only 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 motivation for the introduction of the polar system was the study of circular and orbital motion.
[edit] Position and navigation
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. For instance, aircraft use a slightly modified version of the polar coordinates for navigation. In this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a clockwise direction, rather than counterclockwise, as in the mathematical system. Heading 360 corresponds to magnetic north, while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively.[23] Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read zero-niner-zero by air traffic control).[24]
[edit] 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 use of the polar coordinate system. These systems include gravitational fields, which obey the inverse-square law, as well as systems with point sources, such as radio antennas.
Radially asymmetric systems may also be modeled with polar coordinates. For example, a microphone's pickup pattern illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves. The curve for a standard, the most common unidirectional microphone, can be represented as r = 0.5 + 0.5sin(θ) at its target design frequency.[25] The pattern shifts toward omnidirectionality at lower frequencies.
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Polar Graphs and Microphones
Different microphones have different recording patterns depending on their purpose.
a. Omni-directional Microphone: This microphone is used when we want to record sound from all directions (for example, for a choir).
The recording pattern is almost circular and would correspond to the polar curve r = sin θ that we met above.
omni-directional
Omni-directional microphone [image source]
The following diagram shows a real recording pattern for an omni-directional microphone, graphed on polar graph paper. The different curves are for different frequencies, and the placement of the microphone is at the center of the circle. At low frequencies the pattern is almost circular, but at higher frequencies it becomes less so and more erratic.
omni-directional (real)
Image source
b. Cardioid Microphone: This is a uni-directional microphone, which means we only want to pick up sounds from in front (one direction). The recording pattern is a cardioid, which we met above. In the following image, I have used the graph of r = 1 + sin θ. (There is some “spill”, where sounds immediately behind the microphone are also detected.)
cardioid microphone
Cardioid microphone [image source]
This is a "super-directional" mike, where we only want to pick up sounds from directly in front of the mike. The graph used for this example is r = θ2.
shot-gun mike
Shotgun microphone [image source]
d. Bi-directional Microphone: This is used in an interview situation, where we want to pick up the voices of the interviewer and the person being interviewd.
bi-directional
Bi-directional microphone image: source
So next time you see a microphone (in your mobile phone, notebook computer, in a recording studio or wherever), remember that the shape of its recording pattern is an interesting application of graphs using polar coordinates!
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. For instance, aircraft use a slightly modified version of the polar coordinates for navigation. In this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a clockwise direction, rather than counterclockwise, as in the mathematical system. Heading 360 corresponds to magnetic north, while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively.[23] Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read zero-niner-zero by air traffic control).[24]
[edit] 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 use of the polar coordinate system. These systems include gravitational fields, which obey the inverse-square law, as well as systems with point sources, such as radio antennas.
Radially asymmetric systems may also be modeled with polar coordinates. For example, a microphone's pickup pattern illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves. The curve for a standard, the most common unidirectional microphone, can be represented as r = 0.5 + 0.5sin(θ) at its target design frequency.[25] The pattern shifts toward omnidirectionality at lower frequencies.
--------------------------------------…
Polar Graphs and Microphones
Different microphones have different recording patterns depending on their purpose.
a. Omni-directional Microphone: This microphone is used when we want to record sound from all directions (for example, for a choir).
The recording pattern is almost circular and would correspond to the polar curve r = sin θ that we met above.
omni-directional
Omni-directional microphone [image source]
The following diagram shows a real recording pattern for an omni-directional microphone, graphed on polar graph paper. The different curves are for different frequencies, and the placement of the microphone is at the center of the circle. At low frequencies the pattern is almost circular, but at higher frequencies it becomes less so and more erratic.
omni-directional (real)
Image source
b. Cardioid Microphone: This is a uni-directional microphone, which means we only want to pick up sounds from in front (one direction). The recording pattern is a cardioid, which we met above. In the following image, I have used the graph of r = 1 + sin θ. (There is some “spill”, where sounds immediately behind the microphone are also detected.)
cardioid microphone
Cardioid microphone [image source]
This is a "super-directional" mike, where we only want to pick up sounds from directly in front of the mike. The graph used for this example is r = θ2.
shot-gun mike
Shotgun microphone [image source]
d. Bi-directional Microphone: This is used in an interview situation, where we want to pick up the voices of the interviewer and the person being interviewd.
bi-directional
Bi-directional microphone image: source
So next time you see a microphone (in your mobile phone, notebook computer, in a recording studio or wherever), remember that the shape of its recording pattern is an interesting application of graphs using polar coordinates!
