Envelope (mathematics)

In mathematics, an envelope of a family of manifolds (especially a family of curves) is a manifold that is tangent to each member of the family at some point.

Envelope of a family of curves

The simplest formal expression for an envelope of curves in the $(x,y)$-plane is the pair of equations

: $F(x,y,t)=0qquadqquad(1),$

: $frac\{partial\; F(x,y,t)\}\{partial\; t\}=0qquadqquad(2),$

where the family is implicitly defined by (1). Obviously the family has to be "nicely" — differentiably — indexed by "t".

The logic of this form may not be obvious, but in the vulgar: solutions of (2) are places where $F(x,y,t)$, and thus $(x,y)$, are "constant" in "t" — "ie", where "adjacent" family members intersect, which is another feature of the envelope.

For a family of plane curves given by parametric equations $(x(t,\; p),\; y(t,\; p)),$, the envelope can be found using the equation

: $\{partial\; xoverpartial\; t\}\{partial\; yoverpartial\; p\}\; =\; \{partial\; yoverpartial\; t\}\{partial\; xoverpartial\; p\}$

where variation of the parameter "p" gives the different curves of the family.

Examples

Example 1

In string art it is common to cross-connect two lines of equally spaced pins. What curve is formed?

For simplicity, set the pins on the "x"- and "y"-axes; a non-orthogonal layout is a rotation and scaling away. A general straight-line thread connects the two points (0, "k"−"t") and ("t", 0), where "k" is an arbitrary scaling constant, and the family of lines is generated by varying the parameter "t". From simple geometry, the equation of this straight line is "y" = −("k" − "t")"x"/"t" + "k" − "t". Rearranging and casting in the form "F"("x","y","t") = 0 gives:

:

Now differentiate "F"("x","y","t") with respect to "t" and set the result equal to zero, to get

:

These two equations jointly define the equation of the envelope. From (2) we have "t" = (−"y" + "x" + "k")/2. Substituting this value of "t" into (1) and simplifying gives an equation for the envelope in terms of "x" and "y" only:

: $x^2\; -\; 2xy\; +\; y^2\; -2kx\; -\; 2ky\; +\; k^2\; =\; 0,$

This is the familiar implicit conic section form, in this case a parabola. Parabolae remain parabolae under rotation and scaling; thus the string art forms a parabolic arc ("arc" since only a portion of the full parabola is produced). In this case an anticlockwise rotation through 45° gives the orthogonal parabolic equation "y" = "x"²/("k"√2) + "k"/(2√2). Note that the final step of eliminating "t" may not always be possible to do analytically, depending on the form of "F"("x","y","t").

Example 2

Another example: $(x-u)v\text{'}=(y-v)u\text{'}$ is a tangent of a parametrised curve $(u(t),v(t))$. If we take $F(x,y,t)=(x-u)v\text{'}-(y-v)u\text{'}$ then $F\_t(x,y,t)=xv"-yu"-uv"+vu"$ and $F=F\_t=0$ gives $(x,y)=(u,v)$ when $v"u\text{'}\; e\; u"v\text{'}$. So a curve is the envelope of its own tangents except where its curvature is zero. (This could also be read as a validation of this analytical form.)

Envelope of a family of surfaces

A one-parameter family of surfaces in three-dimensional Euclidean space is given by a set of equations

:$F(x,y,z,a)=0$

depending on a real parameter "a" (see harvtxt|Eisenhart|2004|p=59-61). For example the tangent planes to a surface along a curve in the surface form such a family.

Two surfaces corresponding to different values "a" and "a' " intersect in a common curve defined by

:$F(x,y,z,a)=0,,,\{F(x,y,z,a^prime)-F(x,y,z,a)over\; a^prime\; -a\}=0.$

In the limit as "a' " approaches "a", this curve tends to a curve contained in the surface at "a"

:$F(x,y,z,a)=0,,,\{partial\; Fover\; partial\; a\}(x,y,z,a)=0.$

This curve is called the characteristic of the family at "a". As "a" varies the locus of these characteristic curves defines a surface called the envelope of the family of surfaces.

ee also

* Ruled surface

References

*citation | last = Eisenhart|first= Luther P.|authorlink=Luther Eisenhart | title=A Treatise on the Differential Geometry of Curves and Surfaces | publisher=Dover | year =2004 | id=ISBN 0486438201|url=http://ia310129.us.archive.org/0/items/treatonthediffer00eiserich/treatonthediffer00eiserich.pdf

External links

* [http://mathworld.wolfram.com/Envelope.html Mathworld]
* [http://www.mpi-sb.mpg.de/~shin/Research/CCurve/node7.html]