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Give only two real life examples of polar coordinates.
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Give only two real life examples of Polar Coordinates.
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.
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.[21] 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).[22]
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 cardioid microphone, the most common unidirectional microphone, can be represented as r = 0.5 + 0.5sin(θ) at its target design frequency.[23] The pattern shifts toward omni directionality at lower frequencies.
NOW THAT YOU KNOW A MORE GENERAL SENSE OF WHEN POLAR COORDINATES CAN BE APPLIED, PERHAPS YOU CAN COME UP WITH A FEW OF YOUR OWN EXAMPLES.
two real life examples of polar coordinates:
(1) 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.
(2) Radar:
In radar a transmitter emits a pulse of radio waves which are reflected by the target and detected by the receiver which is usually in the same place as the transmitter. Hence, the receiver gets two pieces of information; the angle from a reference line and the distance. This information is best used in the polar coordinate system.
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 cardioid microphone, 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]
c. Shotgun Microphone: 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!
Physicists and economists use coordinate planes to show the connection between two factors, and geographers use them in mapping. Therefore, the most common applications of Polar coordinates in real life is creating simple graphs to depict information or spotting a specific location on the map.
How Are Polar Coordinates Used in Real Life?
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.
What is an application of polar graphs/ coordinates in the ''real world''?'?
Polar coordinates in the real world can be found in calculations of distance or direction. An application of polar graphs/coordinates in the 'real world' is their use in systems using center points to calculate directions, such as radio antennas. Polar coordinates are also used in navigation.
Radar measures the things it is tracking by the angle and distance from the antenna. This maritime navigation radar shows the ship's position in the center of the display and other ships and land as distance and direction from the ship. (The straight lines attached to the other ships are computerized course projections for those ships.)
The pick up patterns of microphones are shown on polar plots. This plot shows the pattern for one of the most popular directional microphones used for vocal performance. The pattern is called "cardioid," (meaning heart-shaped). You can see that the shape changes a bit depending on the sound's frequency. Each circle on the plot represents a bit less than doubling of the sound reception, so this microphone "hears" things about 10 times better from the front than from the back.
A familiar way to locate a point in the coordinate plane is by specifying its rectangular coordinates (x,y) —that is, by giving its abscissa x and ordinate y relative to given perpendicular axes. In some problems it is more convenient to locate a point by means of its polar coordinates. The polar coordinates give its position relative to a fixed reference point O (the pole) and to a given ray (the polar axis) beginning at O .
For convenience, we begin with a given xy -coordinate system and then take the origin as the pole and the nonnegative x -axis as the polar axis. Given the pole O and the polar axis, the point P with polar coordinates r and θ , written as the ordered pair (r,θ) , is located as follows. First find the terminal side of the angle θ , given in radians, where θ is measured counterclockwise (if θ>0 ) from the x -axis (the polar axis) as its initial side. If r≥0 , then Pis on the terminal side of this angle at the distance r from the origin. If r<0 , then P lies on the terminal side of this angle at the distance r from the origin. If r<0 , then P lies on the ray opposite the terminal side at the distance ∣r∣=-r>0 from the pole. The radial coordinater can be described as the directed distance from P the pole along the terminal side of the angle θ . Thus, if r is positive, the point P lies in the same quadrant as θ , whereas if r is negative, then P lies in the opposite quadrant. If r=0 , the angle θ does not matter; the polar coordinates (0,θ) represent the origin whatever the angular coordinate θ might be. The origin, or pole, is the only point for which r=0 .
(2,π3), (-2,4π3), (2,7π3), (-2,-2π3) |
To convert polar coordinates into rectangular coordinates, we use the basic relations
x=rcosθ, y=rsinθ | (1) |
that we read from the right triangle below.
Converting in the opposite direction, we have
r2=x2+y2, tanθ=yx if x≠0. | (2) |
Some care is required in making the correct choice of θ in the formula tanθ=y/x . If x>0 , then (x,y) lies in either the first or fourth quadrant, so -π/2<θ<π/2 , which is the range of the inverse tangent function. Hence if x>0 , then θ=arctan(y/x) . But if x<0 , then (x,y) lies in the second or third quadrant. In this cae a proper choice for the angle θ=π+arctan(y/x) . In any event, the signs of x and y in Eqs. ( 1 ) with r>0 indicate the quadrant in which θ lies.
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