Envelope

Remarks

Differentiating $F(\gamma(c), c) = 0$ and using clause 3 gives $\partial F / \partial c\, (\gamma(c), c) = 0$, where $\partial F/\partial c$ is the partial derivative in $c$. So the points of an envelope satisfy the two equations $F = 0$ and $\partial F / \partial c = 0$. Eliminating $c$ gives the discriminant set, which contains the envelope but may also contain singular points of the curves. Example: $F(x, y, c) = x \cos c + y \sin c - 1$, the lines at distance $1$ from the origin. Then $\gamma(c) = (\cos c, \sin c)$ satisfies $F = \cos^2 c + \sin^2 c - 1 = 0$, and $\gamma'(c) = (-\sin c, \cos c)$ is orthogonal to $\nabla F = (\cos c, \sin c)$; so the unit circle is the envelope. Non-example: the concentric circles $x^2 + y^2 - c^2 = 0$, $c > 0$, have no envelope, since $\partial F / \partial c = -2c \ne 0$ and clause 3 can never hold together with clause 1. For a regular curve $\beta$, the family of its tangent lines, with parameter $c$ the parameter of $\beta$, has $\beta$ itself as an envelope.