Where does envelope collide?

The single curve (or surface) that is tangent to every member of a family of curves, touching each one in turn and outlining the whole family: as a parameter varies, each curve in the family just grazes this common boundary. Leibniz introduced the construction in his 1694 Nova calculi differentialis applicatio, finding the curve touched by a family defined by a varying condition, and it was later set out as standard differential geometry, for example in Bruce and Giblin's 1992 Curves and singularities. Here the referent is a purely geometric object, a bounding curve computed from a one-parameter family, and not a result about an optimum, an amplitude limit on a signal, or a membrane around a cell nucleus.[1][2]

Unit

A single bounding curve or surface derived from a one-parameter family

Recovery means

Not applicable

Measured by

Found analytically by eliminating the parameter between the family equation and its derivative with respect to that parameter

In microeconomics the envelope is the boundary curve traced by a family of cost or value curves, and the theorem that describes how an optimised value shifts when a parameter changes. Viner's 1931 Cost curves and supply curves drew the long-run average cost curve as the outline that touches each short-run average cost curve from below, one for every scale of plant. Samuelson's 1947 Foundations of economic analysis generalised this into the envelope theorem, which states that the change in an optimised objective with respect to a parameter equals its direct partial change, holding the optimal choice fixed. The referent is an economic result about optima and cost boundaries, distinct from the bare geometric curve, an amplitude limit on a signal, or a membrane around a cell nucleus.[3][4]

Unit

A boundary cost curve, or the rate of change of an optimised value with a parameter

Recovery means

Not applicable

Measured by

Derived from estimated cost or value functions using their tangency and first-order derivative conditions

The envelope of a signal is the smooth, slowly varying curve that outlines the peaks of a faster oscillation, tracing its instantaneous amplitude over time. Gabor's 1946 Theory of communication gave the rigorous definition by pairing a real signal with an analytic companion whose magnitude is the amplitude envelope, which is how a modulating tone is recovered from a carrier wave. The referent is a bounding amplitude curve read off a waveform, and it is unrelated to a geometric envelope of a curve family, an economic optimum, or a membrane around a cell nucleus.[5]

Unit

The instantaneous amplitude curve of an oscillating waveform over time

Recovery means

Not applicable

Measured by

Computed as the magnitude of the analytic signal via the Hilbert transform, or by peak detection and rectified smoothing

The nuclear envelope is the double membrane that encloses the nucleus of a eukaryotic cell, walling the genetic material off from the surrounding cytoplasm and perforated by pores that gate what crosses. Callan and Tomlin's 1950 Experimental studies on amphibian oocyte nuclei used the electron microscope to show that this boundary is a two-layered membrane pierced by regular pores, giving the structure its modern description. The referent is a physical cellular structure, a wrapper of lipid membrane, sharing only the word with a geometric bounding curve, an economic optimum, or a signal's amplitude limit.[6]

Unit

One double membrane surrounding a single cell nucleus

Recovery means

It disassembles and reassembles across cell division as the nucleus breaks down and re-forms

Measured by

Visualised by electron and fluorescence microscopy and characterised by its membrane proteins and pore complexes