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Characteristics Of The Polar Coordinate System

Referencing Styles : MLA | Pages : 1

Within the field of mathematics, the polar coordinate system is a coordinate system which possess the feature of being two-dimensional. Polar coordinate system is determined taking into consideration the distance from the reference point and an angle from a specific reference direction. The reference point which is taken into consideration is referred to as the ‘pole’, and the ray from this pole towards the reference direction is known as the polar axis. The distance starting from the pole is called the radial coordinate, simply radius or the radial distance. The angle is known as the angular coordinate, azimuth or the polar angle. The radial coordinate is more often denoted by r or ρ. On the other hand, the angular coordinate is specified by the symbol φ, θ or t. the angular coordinate is denoted by φ and by ISO standard 31-11. Howsoever, in literature related to mathematics the angle is often specified by θ instead of φ.

Angles within the polar notation is generally expressed with the help of degrees or radians. Degrees are usually associated with navigation, surveying and many other applied disciplines, while radians are more common within the field of mathematics and physics contacting mathematics. The angle φ is programmed to start at 0° from a specified reference direction, and to increase the rotations in either towards the clockwise or counterclockwise orientation. In the field of mathematics, for example, the reference direction is usually marked as a ray from the pole horizontally towards the right, and the polar angle increases towards positive angles for counterclockwise rotations. On the other hand, for the navigation the 0°-heading is drawn vertically upward and after which the angle increases for clockwise rotations. The polar angle decreases to the negative values for rotations within the respective opposite orientations.

Adding any multiple number of full turns (360°) with the angular coordinate does not necessarily change the corresponding direction specified. In the same manner, any polar coordinate which is identical to the coordinate consisting of the negative radial component and in the opposite direction. Hence, the same point can be expressed with the help of an infinite number of different polar coordinates respectively, where n is any arbitrary integer. Places where there is a need of any unique representation for any point beside the pole, it is a usual aspect to limit r to the positive numbers and φ to the interval. Another convention, with reference to the usually considered codomain of the arctan-function, is to make provision for the arbitrary nonzero related real values of the radial component and to place a restriction upon the polar angle. In all of these cases, a unique azimuth for the pole must be chosen.

The equation which defines an algebraic equation expressed with the help of polar coordinates is usually referred to as the polar equation. In most of the cases, such a specific equation can be simple specified by defining r as a function related to φ. The resulting curve then includes the points in the form which was specified. 

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