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Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3, 60°). In blue, the point (4, 210°). In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...
The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question. The azimuthal angle is denoted by φ ∈ [ 0 , 2 π ] {\displaystyle \varphi \in [0,2\pi ]} : it is the angle between the x -axis and the projection of the radial vector onto the xy -plane.
A point in the complex plane can be represented by a complex number written in cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers.
The radius and the azimuth are together called the polar coordinates, as they correspond to a two-dimensional polar coordinate system in the plane through the point, parallel to the reference plane. The third coordinate may be called the height or altitude (if the reference plane is considered horizontal), longitudinal position, [1] or axial ...
Conversely, the polar line (or polar) of a point Q in a circle C is the line L such that its closest point P to the center of the circle is the inversion of Q in C. If a point A lies on the polar line q of another point Q, then Q lies on the polar line a of A. More generally, the polars of all the points on the line q must pass through its pole Q.
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ).
Two-center bipolar coordinates. In mathematics, two-center bipolar coordinates is a coordinate system based on two coordinates which give distances from two fixed centers and . [1] This system is very useful in some scientific applications (e.g. calculating the electric field of a dipole on a plane). [2] [3]