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  2. Polar coordinate system - Wikipedia

    en.wikipedia.org/wiki/Polar_coordinate_system

    Toggle Converting between polar and Cartesian coordinates subsection. ... 4.7 Intersection of two polar curves. 5 ... The formula for the area of R is retrieved by ...

  3. Fermat's spiral - Wikipedia

    en.wikipedia.org/wiki/Fermat's_spiral

    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 ...

  4. Hyperbola - Wikipedia

    en.wikipedia.org/wiki/Hyperbola

    Two tangent lines to B have no (finite) poles because they pass through the center C of the reciprocation circle C; the polars of the corresponding tangent points on B are the asymptotes of the hyperbola. The two branches of the hyperbola correspond to the two parts of the circle B that are separated by these tangent points.

  5. Coordinate systems for the hyperbolic plane - Wikipedia

    en.wikipedia.org/wiki/Coordinate_systems_for_the...

    The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis.

  6. Lemniscate of Bernoulli - Wikipedia

    en.wikipedia.org/wiki/Lemniscate_of_Bernoulli

    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 ...

  7. Archimedean spiral - Wikipedia

    en.wikipedia.org/wiki/Archimedean_spiral

    Equivalently, in polar coordinates (r, θ) it can be described by the equation = with real number b. Changing the parameter b controls the distance between loops. From the above equation, it can thus be stated: position of the particle from point of start is proportional to angle θ as time elapses.

  8. Spherical coordinate system - Wikipedia

    en.wikipedia.org/wiki/Spherical_coordinate_system

    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

  9. Polar curve - Wikipedia

    en.wikipedia.org/wiki/Polar_curve

    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.