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A polar plotter also known as polargraph or Kritzler is a plotter which uses two-center bipolar coordinates to produce vector drawings using a pen suspended from strings connected to two pulleys at the top of the plotting surface. This gives it two degrees of freedom and allows it to scale to fairly large drawings simply by moving the motors ...
Polar coordinates are used often in navigation as the destination or direction of travel can be given as an angle and distance from the object being considered. For instance, aircraft use a slightly modified version of the polar coordinates for navigation. In this system, the one generally used for any sort of navigation, the 0° ray is ...
Vectors are defined in cylindrical coordinates by (ρ, φ, z), where ρ is the length of the vector projected onto the xy-plane, φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by:
Because a polar coordinate plot is limited to polar angles between 0 and 2π, there are 2π / T = k cycles displayed in the graph. No additional points need be plotted because the radial coordinate at θ = 0 is the same value at θ = 2π (which are crests for two different positive half-cycles for roses specified by the cosine function).
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 ).
Another common coordinate system for the plane is the polar coordinate system. [7] A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line).
Log-polar coordinates in the plane consist of a pair of real numbers (ρ,θ), where ρ is the logarithm of the distance between a given point and the origin and θ is the angle between a line of reference (the x-axis) and the line through the origin and the point.
A third choice is the Gaussian polar chart, which correctly represents radial distances, but distorts transverse distances and angles. There are other possible charts; the article on spherically symmetric spacetime describes a coordinate system with intuitively appealing features for studying infalling matter.