Envelope

Let a one-parameter family of curves by given implicitly by

$$\varphi (x,y,C)=0,C\in \mathbb{R}$$

The envelope of this family is a curve $\psi (x,y)=0$ with the property that:

For every point $(x_0,y_0)$ on $\psi$, there exists a parameter value $C_0$ such that:

1. The point $(x_0,y_0)$ lies on the curve defined by $\varphi (x,y,C_0)=0$.
2. The curve $\varphi (x,y,C_0)=0$ is tangent to the envelope $\psi (x,y)=0$ at the point $(x_0,y_0)$.

Analytically, the envelope is obtained by eliminating $C$ from the system

$$\{\begin{array}{l}\varphi (x,y,C)=0\\ \frac{\partial \varphi }{\partial C}(x,y,C)=0\end{array}$$