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The p-th polar of a C for a natural number p is defined as Δ Q p f(x, y, z) = 0. This is a curve of degree n−p. When p is n−1 the p-th polar is a line called the polar line of C with respect to Q. Similarly, when p is n−2 the curve is called the polar conic of C.
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.
The north polar circle on a polar projection. The polar circle as lines on a modified cylindrical projection. The Arctic Circle in Finland, 1975. The Arctic Circle in Norway at Saltfjellet mountain plateau in July 2003. A polar circle is a geographic term for a conditional circular line (arc) referring either to the Arctic Circle or the ...
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).
In mathematics, a rose or rhodonea curve is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates. Rose curves or "rhodonea" were named by the Italian mathematician who studied them, Guido Grandi , between the years 1723 and 1728.
Newton took the origin P inside the oval, and considered the spiral of points (r, θ) in polar coordinates whose distance r from P is the area cut off by the lines from P with angles 0 and θ. He then observed that this spiral cannot be algebraic as it has an infinite number of intersections with a line through P , so the area cut off by a ...
Construction of the limaçon r = 2 + cos(π – θ) with polar coordinates' origin at (x, y) = ( 1 / 2 , 0). In geometry, a limaçon or limacon / ˈ l ɪ m ə s ɒ n /, also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius.