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PR is the diameter of a circle centered on O; its radius AO is the arithmetic mean of a and b. Using the geometric mean theorem, triangle PGR's altitude GQ is the geometric mean. For any ratio a:b, AO ≥ GQ. A semicircle can be used to construct the arithmetic and geometric means of two lengths using straight-edge and compass.
This number is a quartic analog of the = 3.141592..., ratio of perimeter to diameter of a circle. As complex functions, sl and cl have a square period lattice (a multiple of the Gaussian integers) with fundamental periods {(+), ()}, [4] and are a special case of two Jacobi elliptic functions on that lattice, = (;), = (;).
P ' is the inverse of P with respect to the circle. To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P ', lying on the ray from O through P ...
The haversine function computes half a versine of the angle θ, or the squares of half chord of the angle on a unit circle (sphere). To solve for the distance d, apply the archaversine (inverse haversine) to hav(θ) or use the arcsine (inverse sine) function:
The fundamental rectangle in the complex plane of . There are twelve Jacobi elliptic functions denoted by (,), where and are any of the letters , , , and . (Functions of the form (,) are trivially set to unity for notational completeness.) is the argument, and is the parameter, both of which may be complex.
In those days the theory of elliptic functions and the theory of doubly periodic functions were considered to be different theories. They were brought together by Briot and Bouquet in 1856. [20] Gauss discovered many of the properties of elliptic functions 30 years earlier but never published anything on the subject. [21]
The metric of the model on the half-plane, { , >}, is: = + ()where s measures the length along a (possibly curved) line. The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose centers are on the x-axis) and straight vertical rays ...
The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle: For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ) is the distance AC (half of the chord).