Crossing out region on surface

Consider the following Wolfram Mathematica code

Clear["Global*"];

nn          = 3;
eps         = Table[Subscript[ep,i],{i,nn}];
xs          = Table[Subscript[x,i],{i,nn}];
alp         = 1/2;
bet         = 2;
Subscript[ep,2] = -alp Subscript[ep,1]/bet;
Subscript[ep,1] = 1;
Subscript[ep,3] = 0;

eqNcar      = -(1 + alp + bet)^(-1) - ((1 + alp + bet)^(-1) - Subscript[x, 1])^2 - ((1 + alp + bet)^(-1) - Subscript[x, 2])^2 + Subscript[x, 3];
eqNpar      = {w,s,w^2+s^2} + 1/(1+ alp+bet) {1,1,1};

eq1         = (-((1 + (1 + alp + bet)*s^2 + (1 + alp + bet)*w^2)*\
(bet*s + s^2 + w*(alp + w))) + 2*s*(1 + (1 + alp + bet)*s)*\
(s + alp*s^2 + w*(bet + alp*w)) + 2*w*(1 + (1 + alp + bet)*w)*\
(alp*s + w + bet*(s^2 + w^2)))/(1 + alp + bet);

eq2         = (alp*(-s + w)*((1 + alp + bet)^(-1) + s^2 + w^2) -
(2*w*((1 + alp + bet)^(-1) + w)*(-(alp^2*((1 + alp + bet)^(-1) + s)) +
bet*((1 + alp + bet)^(-1) + w)))/bet -
(2*s*((1 + alp + bet)^(-1) + s)*(-(alp*((1 + alp + bet)^(-1) + s)) +
bet^2*((1 + alp + bet)^(-1) + w)))/bet)*Subscript[ep, 1];

pN = ParametricPlot3D[eqNpar,{w,-1,1},{s,-1,1},\
MeshStyle -> {{Blue},{Red}},\
Mesh -> {{0}},PlotStyle->Orange,PlotPoints->200,
BoundaryStyle->{1->None},\
AxesLabel->xs
];

cm              = 72/2.54;
image       = Rasterize[Show[pN,ImageSize->10 cm],"Image",ImageResolution->800];
Export["ex.pdf", Show[image, ImageSize -> 10 cm]]


The cartesian of the surface $$N$$ is eqNcar. I draw the intersections $$N\cap E_1$$ in blue and $$N\cap E_2$$ in red where the cartesian equations of $$E_1$$, $$E_2$$ are eq1 and eq2 respectively.

I would like to add a pattern on a specific region of $$N$$. Namely, is is possible to color or to cross out or to draw white waves in the region : $$N\cap \{eq2<0\} \cap \{eq1>0\}$$ ?

I tried using RegionPlot3D but I did not get acceptable results. I also tried by defining a region with Region but it takes a really long time to process the image (with DiscretizeRegion) and the render was not good.

Is there a way to do this ?

• RegionFunction -> Function[{x, y, z, w, s}, eq2 < 0 && eq1 > 0]? Jul 8, 2023 at 13:58
• Or MeshShading -> {{Automatic, Automatic}, {HalftoneShading[], Automatic}} Jul 8, 2023 at 14:03

MeshShading -> {{Automatic, Automatic}, {HalftoneShading[],     Automatic}}
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