Envelope (mathematics) - Envelope of A Family of Curves

Envelope of A Family of Curves

Let each curve C_t in the family be given by f_t(x, y)=0, where t is a parameter. Write F(t, x, y)=f_t(x, y) and assume F is differentiable.

The envelope of the family C_t 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 C_t and C_u 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 C_t be the line whose x and y intercepts are t and 1−t, this is shown in the animation above. The equation of C_t 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 C_θ 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