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The equation defining a plane curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of φ. The resulting curve then consists of points of the form (r(φ), φ) and can be regarded as the graph of the polar function r.
If the polar line of C with respect to a point Q is a line L, then Q is said to be a pole of L. A given line has (n−1) 2 poles (counting multiplicities etc.) where n is the degree of C. To see this, pick two points P and Q on L. The locus of points whose polar lines pass through P is the first polar of P and this is a curve of degree n−1.
The equations for x and y can be combined to give + = (+) [2] [3] or + = (). This equation shows that σ and τ are the real and imaginary parts of an analytic function of x+iy (with logarithmic branch points at the foci), which in turn proves (by appeal to the general theory of conformal mapping) (the Cauchy-Riemann equations) that these particular curves of σ and τ intersect at ...
A polar diagram could refer to: Polar area diagram, a type of pie chart; Radiation pattern, in antenna theory; A diagram based on polar coordinates; Spherical coordinate system, the three-dimensional form of a polar response curve; In sailing, a Polar diagram is a graph that shows a sailing boats potential wind speed over a range of wind and ...
Archimedean spiral represented on a polar graph. The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance (equal to 2πb if θ is measured in radians), hence the name "arithmetic spiral".
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.
Non-Euclidean geometry often makes appearances in works of science fiction and fantasy. In 1895, H. G. Wells published the short story "The Remarkable Case of Davidson's Eyes". To appreciate this story one should know how antipodal points on a sphere are identified in a model of the elliptic plane. In the story, in the midst of a thunderstorm ...
This is a list of Wikipedia articles about curves used in different fields: mathematics (including geometry, statistics, and applied mathematics), ...