Editing Envelope (mathematics) (section)

==Envelope of a family of curves==
Let each curve ''C''<sub>''t''</sub> in the family be given as the solution of an equation ''f''<sub>''t''</sub>(''x'',&nbsp;''y'')=0 (see [[implicit curve]]), where ''t'' is a parameter. Write ''F''(''t'',&nbsp;''x'',&nbsp;''y'')=''f''<sub>''t''</sub>(''x'',&nbsp;''y'') and assume ''F'' is differentiable.

The envelope of the family ''C''<sub>''t''</sub> is then defined as the set <math>\mathcal{D}</math> of points (''x'',''y'') for which, simultaneously, 
:<math>F(t, x, y) = 0~~\mathsf{and}~~{\partial F \over \partial t}(t, x, y) = 0</math>
for some value of ''t'',
where <math>\partial F/\partial t</math> is the [[partial derivative]] of ''F'' with respect to ''t''.<ref>{{Citation |first=J. W. |last=Bruce |first2=P. J. |last2=Giblin |title=Curves and Singularities |publisher=Cambridge University Press |year=1984 |ISBN=0-521-42999-4}}</ref>

If ''t'' and ''u'', ''t''≠''u'' are two values of the parameter then the intersection of the curves ''C''<sub>''t''</sub> and ''C''<sub>''u''</sub> is given by 
:<math>F(t, x, y) = F(u, x, y) = 0\,</math>
or, equivalently,
:<math>F(t, x, y) = 0~~\mathsf{and}~~\frac{F(u, x, y)-F(t, x, y)}{u-t} = 0.</math>
Letting ''u'' → ''t'' gives the definition above.

An important special case is when ''F''(''t'',&nbsp;''x'',&nbsp;''y'') is a polynomial in ''t''. This includes, by [[clearing denominators]], the case where ''F''(''t'',&nbsp;''x'',&nbsp;''y'') is a rational function in ''t''. In this case, the definition amounts to ''t'' being a double root of ''F''(''t'',&nbsp;''x'',&nbsp;''y''), so the equation of the envelope can be found by setting the [[discriminant]] of ''F'' to 0 (because the definition demands F=0 at some t and first derivative =0 i.e. its value 0 and it is min/max at that t).

For example, let ''C''<sub>''t''</sub> be the line whose ''x'' and ''y'' intercepts are ''t'' and 11−''t'', this is shown in the animation above. The equation of ''C''<sub>''t''</sub> is
:<math>\frac{x}{t}+\frac{y}{11-t}=1</math>
or, clearing fractions,
:<math>x(11-t)+yt-t(11-t)=t^2+(-x+y-11)t+11x=0.\,</math>
The equation of the envelope is then
:<math>(-x+y-11)^2-44x=(x-y)^2-22(x+y)+121=0.\,</math>

Often when ''F'' is not a rational function of the parameter it may be reduced to this case by an appropriate substitution. For example, if the family is given by ''C''<sub>θ</sub> with an equation of the form ''u''(''x'',&nbsp;''y'')cos θ+''v''(''x'',&nbsp;''y'')sin θ=''w''(''x'',&nbsp;''y''), then putting ''t''=''e''<sup>''i''θ</sup>, cos θ=(''t''+1/''t'')/2, sin θ=(''t''-1/''t'')/2''i'' changes the equation of the curve to
:<math>u{1 \over 2}(t+{1\over t})+v{1 \over 2i}(t-{1\over t})=w</math>
or
:<math>(u-iv)t^2-2wt+(u+iv)=0.\,</math>
The equation of the envelope is then given by setting the discriminant to 0:
:<math>(u-iv)(u+iv)-w^2=0\,</math>
or 
:<math>u^2+v^2=w^2.\,</math>

===Alternative definitions===

# The envelope ''E''<sub>1</sub> is the limit of intersections of nearby curves ''C''<sub>''t''</sub>.
# The envelope ''E''<sub>2</sub> is a curve tangent to all of the ''C''<sub>''t''</sub>.
# The envelope ''E''<sub>3</sub> is the boundary of the region filled by the curves ''C''<sub>''t''</sub>.

Then <math>E_1 \subseteq \mathcal{D}</math>, <math>E_2 \subseteq \mathcal{D}</math> and <math>E_3 \subseteq \mathcal{D}</math>, where <math>\mathcal{D}</math> is the set of points defined at the beginning of this subsection's parent section.