Envelope (mathematics)

In mathematics envelope referred to ( according to French enveloppe, wrapping, also envelope or envelope ) is a curve that envelops a family of curves. That is, the envelope curve touches each flock once. Envelopes arise among other moving objects, such as opening and closing a garage door. Each plane curve is the envelope of its tangents.

The evolute E of a plane curve C is envelope of its normals. C is then the involute of E.

Definition

A graph H is an envelope of curves if the following conditions are met:

Calculation of wrapper functions

Examples

Three-dimensional envelope

Consider the parameterized and by the equation

Defined straight lines.

As described above, the envelope of this family of straight lines is given by the equations

Given. Elimination of supplies nonparametric representation of the envelope:

Trajectories

Another example is the envelope of trajectories. Details are given below envelope parabolic trajectory.

Application

Envelopes are well suited to describe the required space for moving objects. So you can find with envelopes, if you get a wardrobe around a corner in the hallway, or how narrow a street may be in a curve, and how to produce something that a truck can drive safely on her. For most engineering applications, numerical methods are best suited.

In economics speak with him about the time-varying cost functions also of upper and lower -envelope. That between these two is the whole range of cost curves, or differently at any time is realized in the upper and lower envelopes of the true cost function.

Envelope of surfaces

Surfaces can also be described as the envelope of surface droves. for example:

• Channel and pipe surfaces are envelopes of the ball droves.
• Slope surfaces are the envelope of cone droves.