Where does quadrature collide?

The computation of an area or, more generally, of a definite integral: historically to square a figure meant to construct a square of equal area, and the word came to name any method that finds such an area or integral, including numerical rules that approximate it from sampled values. Archimedes's Quadrature of the parabola (ca. 3rd century BCE) found the exact area of a parabolic segment by summing an infinite series, an early instance of the idea. Davis and Rabinowitz's 1984 Methods of numerical integration collected the modern rules (Newton-Cotes, Gaussian, adaptive) that estimate an integral to a chosen accuracy. The referent is a number, an area or integral produced by calculation, not a 90-degree configuration of a planet or a 90-degree offset between two signals.[1][2]

Unit

One computed value: an area or definite integral, in the units of the quantity integrated

Recovery means

Not applicable

Measured by

Obtained by evaluating a quadrature rule (for example Gaussian or Newton-Cotes) on sampled values and bounding its error

The configuration in which a body, seen from Earth, lies 90 degrees away from the Sun along the sky: the Moon is at quadrature at its first and last quarter, and a superior planet is at quadrature when the Sun-Earth-body angle is a right angle. Ptolemy's Almagest (ca. 150 CE) laid out the geometry of such solar elongations within its planetary models, and modern handbooks such as Meeus's 1998 Astronomical algorithms give the formulae for computing the exact times. The referent is a geometric arrangement of the Sun, Earth, and a third body, marked in degrees of elongation, not an area found by calculation or a phase relationship inside an electronic signal.[3][4]

Unit

One geometric configuration: the moment a body's elongation from the Sun equals 90 degrees

Recovery means

Not applicable

Measured by

Computed from ephemerides, or observed as the elongation angle between the body and the Sun reaches a right angle

Two signals, or two components of one signal, that differ in phase by 90 degrees, so that when one is at its peak the other is at zero: such a pair, labelled the in-phase (I) and quadrature (Q) components, can jointly carry the amplitude and phase of a waveform. Gabor's 1946 Theory of communication introduced the complex, or analytic, representation whose imaginary part is the 90-degree-shifted companion of a real signal, and later communication texts such as Haykin's 2001 Communication systems built quadrature modulation and demodulation on that pair. The referent is a phase offset between electrical or sampled signals, measured in degrees or radians, not an integral computed as an area or a planet standing at a right angle to the Sun.[5][6]

Unit

One phase relationship: a 90-degree offset between two signals or between the I and Q components

Recovery means

Not applicable

Measured by

Recovered by mixing a signal with a reference and its 90-degree-shifted copy, or via the Hilbert transform, to separate the I and Q components