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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 the point's distance from a reference point called the pole, and; the point's direction from the pole relative to the direction of the polar axis, a ray drawn from the pole.
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.
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 ...
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are the radial distance r along the line connecting the point to a fixed point called the origin; the polar angle θ between this radial line and a given polar axis; [a] and
For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value |z| of the corresponding z is the amplitude and the argument arg z is the phase.
Lines in a Cartesian plane, or more generally, in affine coordinates, can be described algebraically by linear equations. In two dimensions, the equation for non-vertical lines is often given in the slope-intercept form: = + where: m is the slope or gradient of the line. b is the y-intercept of the line.
In this polar decomposition, the unit circle has been replaced by the line x = 1, the polar angle by the slope y/x, and the radius x is negative in the left half-plane. If x 2 ≠ y 2, then the unit hyperbola x 2 − y 2 = 1 and its conjugate x 2 − y 2 = −1 can be used to form a polar decomposition based on the branch of the unit hyperbola ...
The Fermat spiral with polar equation = can be converted to the Cartesian coordinates (x, y) by using the standard conversion formulas x = r cos φ and y = r sin φ.Using the polar equation for the spiral to eliminate r from these conversions produces parametric equations for one branch of the curve: