Quadrature

As π is transcendental, the quadrature of a circle - a classical problem in geometry - is impossible.

The story that Anaxagoras wrote a treatise on perspective as applied to scene-painting is most improbable; and the statement that he composed a mathematical work dealing with the quadrature of the circle is due to misunderstanding of an expression in Plutarch.

1976, D. T. Whiteside (editor), The Mathematical Papers of Isaac Newton, Volume 7: 1691-1695, Cambridge University Press, page 3, Craige was then on the point of publishing a short tract wherein he expounded a systematic (if far from general) 'Method of determining the quadrature of figures comprehended by straight lines and curves', there gathering a variety of techniques of rational algebraic quadrature and arc-rectification […] .

1992, Bernard Bielecki, SINC Quadratures for Cauchy Principal Value Integrals, T.O. Espelid, Alan Genz, Springer, Numerical Integration: Recent Developments, Software and Applications, page 81, Three types of SINC quadratures are surveyed for the evaluation of Cauchy principal value integrals ∫_ΓF(t)dt/(t-x), x∈Γ, where Γ is an arc in the complex plane. Under suitable assumptions on F, the quadrature errors are of order O(e), where N is the number of quadrature nodes and c is a constant independent of N.

In this chapter, we present some example applications of the Gaussian quadrature methods discussed in the previous chapters.[…]Figure 4.1 shows the accuracy of the polynomial Gaussian quadratures used to evaluate the first two integrals as a function of N.

The integral is thus given by a variant of the Fejér quadrature with two points at the interval boundaries, namely x₁ = −1 and x_N = 1 analogous to the Lobatto quadratures of the next section.

When the Moon is in quadrature, it appears in the sky as a half-moon.

If ν_(IN) has 50% duty-cycle, then the outputs are in quadrature.

The total mmf in a synchronous machine may be split up into two components—one along the pole axis or direct axis and the other at right angles to this or the quadrature axis.[…]In the case of salient pole construction, however, the reluctances of the magnetic circuits on which the mmfs act are different along the direct axis and the quadrature axis.

Quadrature LO generation is critical to the operation of direct-downconversion receivers [7]. Two common techniques for quadrature generation are divide-by-two frequency dividers, and polyphase filters.