If a plane curve is erased after a large number of its tangents have been drawn, the resulting figure "retains a memory" of the original curve as the envelope of all the lines. More generally, for a one-parameter family of curves with equation f (x, y, λ\lambda) = 0, the envelope is a curve tangent to every curve in the family. Determining envelopes is one of the major problems of differential geometry and amounts to solving the system:
{f(x,y,λ)=0 ∂f∂λ(x,y,λ)=0\left\{\begin{array}{l}f(x,y,\lambda)=0\ \frac{\partial f}{\partial\lambda}(x,y,\lambda)=0\end{array}\right.
In the general case, solving it yields a parametric solution
{x=u(λ) y=v(λ)\left\{\begin{array}{l}x=u(\lambda)\ y=v(\lambda)\end{array}\right.
and, if the parameter λ\lambda can be eliminated, an implicit solution g (x, y) = 0.