Editing Envelope (mathematics) (section)

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

:<math>F(x,y,z,a)=0</math>

depending on a real parameter ''a''.<ref>{{Citation |first=Luther P. |last=Eisenhart |title=A Treatise on the Differential Geometry of Curves and Surfaces |publisher=Schwarz Press |year=2008 |ISBN=1-4437-3160-9}}</ref> 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

:<math> F(x,y,z,a)=0,\,\,{F(x,y,z,a^\prime)-F(x,y,z,a)\over a^\prime -a}=0.</math>

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

:<math> F(x,y,z,a)=0,\,\,{\partial F\over \partial a}(x,y,z,a)=0.</math>

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.

{{quotation|The envelope of a family of surfaces is tangent to each surface in the family along the characteristic curve in that surface.}}