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I am trying to make this tikzpicture using the answer from Drawing manifolds in tikzbut I can not make the intersection of U and V gray. If I make U and V white then the intersection would be white too. My question is how can I make the intersection gray and keeping everything else white ?
\documentclass[border=5mm,tikz]{standalone}\usepackage{amssymb}\usepackage{pgfplots}\usepgfplotslibrary{patchplots}\usetikzlibrary{patterns, positioning, arrows}\pgfplotsset{compat=1.15}\begin{document}\begin{figure}[h]\centering\begin{tikzpicture}% Functions i\path[->] (0.8, 0) edge [bend right] node[left, xshift=-2mm] {$\varphi$} (-1, -2.9);\draw[white,fill=white] (0.06,-0.57) circle (.15cm);% Functions j\path[->] (4.2, 0) edge [bend left] node[right, xshift=2mm] {$\psi$} (6.2, -2.8);\draw[white, fill=white] (4.54,-0.12) circle (.15cm);% Manifold\draw[smooth cycle] plot coordinates{(2,2) (-0.5,0) (3,-2) (5,1)} node at (3,2.3) {$M$};% Help lines%\draw[help lines] (-3,-6) grid (8,6);% Subsets\draw[smooth cycle, dashed] plot coordinates {(1,0) (1.5, 1.2) (2.5,1.3) (2.6, 0.4)} node [label={[label distance=-0.3cm, xshift=-2cm, fill=white]:$U$}] {};\draw[smooth cycle, dashed] plot coordinates {(4, 0) (3.7, 0.8) (3.0, 1.2) (2.5, 1.2) (2.2, 0.8) (2.3, 0.5) (2.6, 0.3) (3.5, 0.0)} node [label={[label distance=-0.8cm, xshift=.75cm, yshift=1cm, fill=white]:$V$}] {};% First Axis\draw[thick, ->] (-3,-5) -- (0, -5) {};\draw[thick, ->] (-3,-5) -- (-3, -2) {};% Arrow from i to j\path[->] (0, -3.85) edge [bend left] node[midway, above]{$\psi_{ij}$} (4.5, -3.85);% Second Axis\draw[thick, ->] (5, -5) -- (8, -5) node [label=above:$\phi_j(U_j)$] {};\draw[thick, ->] (5, -5) -- (5, -2) node [label=right:$\mathbb{R}^m$] {};% Sets in R^m\draw[white] (-0.67, -3.06) -- +(180:0.8) arc (180:270:0.8);\fill[even odd rule, white] [smooth cycle] plot coordinates{(-2, -4.5) (-2, -3.2) (-0.8, -3.2) (-0.8, -4.5)} (-0.67, -3.06) -- +(180:0.8) arc (180:270:0.8);\draw[smooth cycle, dashed] plot coordinates{(-2, -4.5) (-2, -3.2) (-0.8, -3.2) (-0.8, -4.5)};\draw [dashed] (-1.45, -3.06) arc (180:270:0.8);\draw[white] (5.7, -3.06) -- +(-90:0.8) arc (-90:0:0.8);\fill[even odd rule, white] [smooth cycle] plot coordinates{(7, -4.5) (7, -3.2) (5.8, -3.2) (5.8, -4.5)} (5.7, -3.06) -- +(-90:0.8) arc (-90:0:0.8);\draw[smooth cycle] plot coordinates{(7, -4.5) (7, -3.2) (5.8, -3.2) (5.8, -4.5)};\draw (5.69, -3.85) arc (-90:0:0.8);\end{tikzpicture}\end{figure}\end{document}