Definition of $\epsilon$-upper envelope of a function

Let $ u$ be a continuous function in an open set $ \Omega\subset \mathbb{R}^{n}$ , and let $ H$ an open set such that $ \bar{H}\subset \Omega$ . We define , for $ \epsilon >0$ , the upper $ \epsilon$ -envelope of $ u$ (with respect to $ H$ ):

$ u^{\epsilon}(x_{0})=\sup_{x \in \bar{H}}\Big\{u(x)+\epsilon -\frac{1}{\epsilon}|x-x_{0}|^{2}\Big\}$

For $ x_{0} \in H$ . The autor says: the graph of $ u^{\epsilon}$ is the envelope of the graphs family $ \{P^{\epsilon}_{x}\}_{x \in \bar{H}}$ of concave paraboloids of opening $ 2/\epsilon$ and vertex $ (x,u(x)+\epsilon)$ .

First, whats is the envelope of the family of graphs? I considered the one dimensional case where $ u(x)=x$ , $ x\in \mathbb{R}$ , the concave “paraboloid when $ x_{0}=1$ , for example is $ P^{\epsilon}(x)=-\epsilon.(x-1)^{2}+\epsilon +x$ . I used a program and draw the graph, I did vary the value of Epsilon, and I saw that the paraboloid is gradually crossing the whole line. But, but i still do not understand the geometry of that definition