Polar Coordinates: Difference between revisions
(Created page with "== Introduction == Polar coordinates are a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the po...") |
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In a polar coordinate system, each point in the plane is represented by its distance r from the origin and its angle θ from the polar axis. The polar coordinates r and θ can be converted to the Cartesian coordinates x and y by using the trigonometric functions sine and cosine. | In a polar coordinate system, each point in the plane is represented by its distance r from the origin and its angle θ from the polar axis. The polar coordinates r and θ can be converted to the Cartesian coordinates x and y by using the trigonometric functions sine and cosine. | ||
[[Image:Detail-78947.jpg|thumb|center|A photograph of a plane with a point marked. The point is connected to the origin by a line, representing the radius. An angle is marked between the line and the x-axis, representing the polar angle.|class=only_on_mobile]] | |||
[[Image:Detail-78948.jpg|thumb|center|A photograph of a plane with a point marked. The point is connected to the origin by a line, representing the radius. An angle is marked between the line and the x-axis, representing the polar angle.|class=only_on_desktop]] | |||
== Conversion between Cartesian and Polar Coordinates == | == Conversion between Cartesian and Polar Coordinates == | ||
Latest revision as of 07:48, 16 May 2024
Introduction
Polar coordinates are a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the polar angle or azimuth.
History
The concepts of angle and radius were used by ancient peoples of the first millennium BC. The Greeks, for example, had a concept of radius and angle, although no coordinate system was associated with this concept. In On Spirals, Archimedes describes the Spiral of Archimedes, a function whose radius depends on the angle.
Definition
In a polar coordinate system, each point in the plane is represented by its distance r from the origin and its angle θ from the polar axis. The polar coordinates r and θ can be converted to the Cartesian coordinates x and y by using the trigonometric functions sine and cosine.
Conversion between Cartesian and Polar Coordinates
The conversion between Cartesian and Polar coordinates is a fundamental aspect of using polar coordinates. The conversion from Cartesian to Polar coordinates involves finding the radius r and the angle θ from the Cartesian coordinates (x, y). The radius r is given by the Pythagorean theorem as r = √(x² + y²), and the angle θ is given by the inverse tangent as θ = atan(y/x).
Conversely, the conversion from Polar to Cartesian coordinates involves finding the Cartesian coordinates (x, y) from the radius r and the angle θ. The x-coordinate is given by x = r cos θ, and the y-coordinate is given by y = r sin θ.
Applications
Polar coordinates are used in a variety of fields including physics, engineering, and mathematics. In physics, polar coordinates are used in kinematics to describe the motion of objects in two dimensions. In engineering, they are used in navigation systems, such as GPS, to give directions. In mathematics, they are used in complex analysis, a branch of mathematics that investigates functions of complex numbers.