Polar coordinates use an angle and distance to locate a point, while rectangular coordinates use distances. They can be converted using trigonometric formulas. Polar coordinates are useful for circular geometry and modeling real-world phenomena with rounded shapes. They can also be used in three dimensions, creating a spherical coordinate system. Spherical coordinates are used in mapping the Earth.
Polar coordinates are a form of expressing position on a two-dimensional plane. Cartesian coordinates, also called rectangular coordinates, use a distance in each of two dimensions to locate a point, but polar coordinates use an angle and a distance. Distance is sometimes referred to as radius.
Rectangular coordinates are typically denoted (x,y), where x and y are distances along their respective axes. Similarly, polar coordinates are expressed as (r,θ). The letter r is the distance from the origin to the angle represented by the Greek letter theta, θ, where r can be a positive or negative number. If a negative distance is used, the magnitude of the distance does not change, but the direction is taken in the opposite direction to the angle on the other side of the origin. A point in a polar coordinate system may be referred to as representing a vector, with a magnitude of r, a direction of, and a sense of direction, which is the sign of r.
The translation between rectangular and polar coordinates can be achieved through the use of trigonometric formulas. For the conversion from rectangular to polar the following formulas can be applied: θ = tan-1(y/x) and r = √(x2 + y2). For polar to rectangular changes, these equations can be used: x = rcosθ and y = rsinθ.
Polar coordinates tend to be used for any situation where rectangular coordinates would prove difficult or awkward to use, and vice versa. Any application involving circular geometry or radial motion is ideal for polar coordinates, because these geometries can be described with relatively simple equations in a polar coordinate system; their graphs have a more curvilinear or circular appearance than those on rectangular coordinate systems. Consequently, polar coordinates have a use of representing models of real-world phenomena that have similarly rounded shapes.
The applications of polar coordinates are quite varied. Polar coordinate plots have been used to model the sound fields produced by different speaker locations or areas where different types of microphones can pick up sound best. Polar coordinates are of great importance for modeling orbital motions in astronomy and space travel. They are also the graphical basis for the famous Euler Formula, which is regularly applied in mathematics for representing and manipulating complex numbers.
Like their rectangular counterparts, polar coordinates don’t have to be limited to just two dimensions. To express values in three dimensions, a second angle represented by the Greek letter phi, φ, can be added to the coordinate system. Any point can thus be located from the origin by a fixed distance and two angles, and the coordinates (r,θ,φ) can be assigned to it. When this type of nomenclature is used to draw and locate points in three-dimensional space, the coordinate system is designated as a spherical coordinate system. This type of geometry is sometimes referred to as using polar spherical coordinates.
Spherical coordinates actually have a well-known application: they are used in mapping the Earth. The angle is typically latitude and is bounded between minus-90 degrees and 90 degrees, while the angle φ is longitude and is kept between minus-180 and 180 degrees. In this application, r can sometimes be ignored, but it is more often employed for expressing elevation above mean sea level.
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