Envelope

An envelope is a common packaging item, usually made of thin flat material. It is designed to contain a flat object, such as a letter or card.

Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a short-arm cross, or a kite. These shapes allow for the creation of the envelope structure by folding the sheet sides around a central rectangular area. In this manner, a rectangle-faced enclosure is formed with an arrangement of four flaps on the reverse side.

Overview

When the folding sequence is such that the last flap to be closed is on a short side it is referred to in commercial envelope manufacture as a pocket - a format frequently employed in the packaging of small quantities of seeds. Although in principle the flaps can be held in place by securing the topmost flap at a single point (for example with a wax seal), generally they are pasted or gummed together at the overlaps. They are most commonly used for enclosing and sending mail (letters) through a prepaid-postage postal system.

Envelope (mathematics)

In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point. Classically, a point on the envelope can be thought of as the intersection of two "adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.

Envelope of a family of curves

Let each curve C_t in the family be given as the solution of an equation f_t(x, y)=0 (see implicit curve), where t is a parameter. Write F(t, x, y)=f_t(x, y) and assume F is differentiable.

The envelope of the family C_t is then defined as the set of points for which

for some value of t, where is the partial derivative of F with respect to t.

Note that if t and u, t≠u are two values of the parameter then the intersection of the curves C_t and C_u is given by

or equivalently

Letting u→t gives the definition above.

An important special case is when F(t, x, y) is a polynomial in t. This includes, by clearing denominators, the case where F(t, x, y) is a rational function in t. In this case, the definition amounts to t being a double root of F(t, x, y), so the equation of the envelope can be found by setting the discriminant of F to 0.

Envelope (disambiguation)

An envelope is the paper container used to hold a letter being sent by post.

Envelope may also refer to:

In mathematics:

In science:

In applied science: