# envelope

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## envelope

**1.**

*Biology*any enclosing structure, such as a membrane, shell, or skin

**2.**the bag enclosing the gas in a balloon

**3.**

*Maths*a curve or surface that is tangent to each one of a group of curves or surfaces

**4.**

*Electronics*the sealed glass or metal housing of a valve, electric light, etc.

**5.**

*Telecomm*the outer shape of a modulated wave, formed by the peaks of successive cycles of the carrier wave

## Envelope

## Envelope

of a family of curves in a plane (or of surfaces in space), a curve (or surface) that touches at each of its points a single curve (or surface) of the family and is geometrically different from that curve (or surface) in an arbitrarily small neighborhood of the point of contact. The equation of the envelope of a plane family of curves defined by the equation *f(x, y, C*) = 0, which contains a parameter *C*, can be obtained by eliminating *C* from the system of equations

Here it is assumed that *f(x, y, C*) has continuous partial derivatives of the first order with respect to all three arguments. In general, this elimination yields not only the envelope but also the locus of the singular points of the curves of the family, that is, the points for which *f″ _{x}* and

*f″*, vanish simultaneously.

_{y}The following are examples in the plane. (1) The family of circles of radius *R* whose centers lie on a line has as an envelope a pair of lines parallel to the line of the centers and located at a distance *R* from that line (see Figure 1). (2) Any curve is the envelope of the family of its tangents and of the family of its circles of curvature. (3) The envelope of the family of normals to a given curve is its evolute (the evolute of an ellipse is shown in Figure 2).

The envelope of a family of surfaces in space may touch each member of the family at a point or along a curve.

For example, (1) the envelope of a family of spheres of radius *R* with centers lying on a single line is a circular cylinder of radius *R* whose axis is that line (the cylinder touches each sphere along a circle); (2) the envelope of a family of spheres of radius *R* whose centers lie in a single plane is a pair of planes parallel to the plane of the centers and located at a distance *R* from that plane (the planes forming the envelope touch each sphere in a point).

The concept of an envelope has significance not only in geometry but also in certain problems in mathematical analysis (singular solutions in the theory of differential equations) and theoretical physics (the caustic and the wave front in optics).

### REFERENCES

Tolstov, G. P. “K otyskaniiu ogibaiushchei semeistva ploskikh krivykh.”*Uspekhi matematicheskikh nauk*, 1952, vol. 7, no. 4.

La Vallée Poussin, C. J. de.

*Kurs analiza beskonechno malykh*, vol. 2. Leningrad-Moscow, 1933. (Translated from French.)

Il’in, V. A., and E. G. Pozniak.

*Osnovy matematicheskogo analiza*, 3rd ed., part 1. Moscow, 1971.

## envelope

[′en·və‚lōp]## envelope

**1.**The imaginary shape of a building indicating its maximum volume; used to check the plan and setback (and similar restrictions) with respect to zoning regulations.

**2.**The folded-over, continuous edge formed by turning the lowest ply of a built-up roofing membrane over the top surface layer; prevents bitumen from dripping through the exposed edge joints and seepage of water into the insulation.

## envelope

**(1)**A range of frequencies for a particular operation.

**(2)**A group of bits or items that is packaged and treated as a single unit.

**(3)**See also pushing the envelope.