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A curve on the Cartesian plane can be mapped into polar coordinates. In this animation, = + is mapped onto = +. Click on image for details. The equation defining a plane curve expressed in polar coordinates is known as a polar equation.
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 ...
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.
In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry . Analytic geometry is used in physics and engineering , and also in aviation , rocketry , space science , and spaceflight .
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 ).
The point furthest from the pole on the inner loop has the coordinates (,), and on the polar axis, is one-third of the distance from the pole compared to the furthest point of the outer loop. The outer and inner loops intersect at the pole.
Polar motion in arc-seconds as function of time in days (0.1 arcsec ≈ 3 meters). [1] Polar motion of the Earth is the motion of the Earth's rotational axis relative to its crust. [2]: 1 This is measured with respect to a reference frame in which the solid Earth is fixed (a so-called Earth-centered, Earth-fixed or ECEF reference frame). This ...
An animated construction gives an idea of the complexity of the curve (Click for enlarged version). The curve is given by the following parametric equations : [ 2 ] x = sin t ( e cos t − 2 cos 4 t − sin 5 ( t 12 ) ) {\displaystyle x=\sin t\!\left(e^{\cos t}-2\cos 4t-\sin ^{5}\!{\Big (}{t \over 12}{\Big )}\right)}