Envelope of A Family of Curves
Let each curve Ct in the family be given by ft(x, y)=0, where t is a parameter. Write F(t, x, y)=ft(x, y) and assume F is differentiable.
The envelope of the family Ct 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 Ct and Cu 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.
For example, let Ct be the line whose x and y intercepts are t and 1−t, this is shown in the animation above. The equation of Ct is
or, clearing fractions,
The equation of the envelope is then
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 Cθ with an equation of the form u(x, y)cosθ+v(x, y)sinθ=w(x, y), then putting t=eiθ, cosθ=(t+1/t)/2, sinθ=(t-1/t)/2i changes the equation of the curve to
or
The equation of the envelope is then given by setting the discriminant to 0:
or
Read more about this topic: Envelope (mathematics)
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