Reviewing this and that resource, I see that a one parameter family of curves has equation
$$f(x,y,z,t)=0 \tag{1}$$
where $t$ is the parameter. And I see that the equation of the envelope of the family, if it exists, is found by differentiating $(1)$ w.r.t. $t$, setting this equal to $0$, i.e.,
$$\frac{\partial f}{\partial t} = 0$$
then solving this for $t$, and substituting the expression into $(1)$.
My question is what happens when the derivative of $(1)$ w.r.t. $t$ equals a constant? Even more specifically, what happens when $f(x,y,z,t)$ has the form
$$f(x,y,z,t) = t - \phi(x,y,z)$$
?
In this case, of course, the derivative of $f$ w.r.t. $t$ equals $1$, which leaves us without an equation for $t$ that could then be substituted into $f=0$ to solve for the envelope. Does this mean that no envelope can exist for such a function?