Does anybody know how to compute the envelope of a family of curves given in complex form $$F:\mathbb {R} \times \mathbb{R} \rightarrow \mathbb{C}$$ $$(w,c) \mapsto F(w,c)$$
without decomposing $F(w,c)$ into its real and imaginary components, $x(w,c)$ and $y(w,c)$?
I want to work entirely on the complex domain, and not see this problem as finding an envelope of a parametric family of curves
$$G:\mathbb{R} \rightarrow \mathbb{R} \times \mathbb{R}$$ $$w \mapsto G(w,c).$$
For example, given the following family of circles: $F(w,c) = e^{iw}+c\sqrt{2}e^{i*pi/4},$ I know it has two parallel lines as its envelopes. In complex form, they are: $(1-i)z-(1+i) \bar z +- 2i \sqrt{2} = 0.$
How can this be computed entirely in the complex domain without decomposing $F(w,c)$ into $x(w,c)$ and $y(w,c)$, which are the parametric equations of the given family of circles?
Does anybody know a good reference on how to compute the envelope entirely on the complex plane?
Tkanks a lot.