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Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the polar rose , Archimedean spiral , lemniscate , limaçon , and cardioid .
Let φ 1 = 0, φ 2 = 2π; then the area of the black region (see diagram) is A 0 = a 2 π 2, which is half of the area of the circle K 0 with radius r(2π). The regions between neighboring curves (white, blue, yellow) have the same area A = 2a 2 π 2. Hence: The area between two arcs of the spiral after a full turn equals the area of the circle ...
A hyperbola is an open curve with two ... be twice the area between the axis and a ray ... and lines generated by a conic is called pole-polar relation or just ...
The Lobachevsky coordinates are useful for integration for length of curves [2] and area between lines and curves. [example needed] Lobachevsky coordinates are named after Nikolai Lobachevsky one of the discoverers of hyperbolic geometry. Circles about the origin of radius 1, 5 and 10 in the Lobachevsky hyperbolic coordinates.
The area enclosed by the lemniscate is a 2 = 2c 2. The lemniscate is the circle inversion of a hyperbola and vice versa. The two tangents at the midpoint O are perpendicular, and each of them forms an angle of π / 4 with the line connecting F 1 and F 2. The planar cross-section of a standard torus tangent to its inner equator is a ...
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
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.
Scroll compressors, used for compressing gases, have rotors that can be made from two interleaved Archimedean spirals, involutes of a circle of the same size that almost resemble Archimedean spirals, [4] or hybrid curves. Archimedean spirals can be found in spiral antenna, which can be operated over a wide range of frequencies.