In[]:=
In[]:=
Function{a.,c.},Function{u.,v.},a.
1+
Cos[v.],a.2
u.
1+
Sin[v.],c.u.[a,c]2
u.
Out[]=
Function{u.$,v.$},a
1+
Cos[v.$],a2
u.$
1+
Sin[v.$],cu.$2
u.$
Function{u.$,v.$},a
1+
Cos[v.$],a2
u.$
1+
Sin[v.$],cu.$2
u.$
Manipulate[Show[Graphics3D[{GrayLevel[.5],Cylinder[{{0,-100,0},{0,100,0}},.03],Cylinder[{{-100,0,0},{100,0,0}},.03],Cylinder[{{0,0,-100},{0,0,100}},.03]}],ContourPlot3D[x^2/a^2+y^2/b^2-z^2/c^2,{x,-4,4},{y,-4,4},{z,-3,3},Contours->{1},MeshStyle->Gray,ContourStyle->{{Orange,Specularity[White,20]}},PlotPoints->10,MaxRecursion->2],Background->Black,Boxed->False,Axes->None,Lighting->{{"Ambient",GrayLevel[.5]},{"Directional",White,ImageScaled[{1,1,1}]}},ImageSize->{500,357},PlotRange->{{-4,4},{-4,4},{-4,4}},ImagePadding->{{0,0},{0,40}}],"lengths of semi-axes",{{a,1.5,"a"},1,3,Appearance->"Labeled"},{{b,1.5,"b"},1,3,Appearance->"Labeled"},{{c,2,"c"},1,3,Appearance->"Labeled"}]
https://demonstrations.wolfram.com/ModelingTheHyperboloidOfOneSheet/
https://demonstrations.wolfram.com/ModelingTheHyperboloidOfOneSheet/
m1={{{0,0},{11.150507`6.,4.116821`6.},{-0.174684`6.,7.725432`6.},{0,0}},{{-0.174684`6.,7.725432`6.},{11.150507`6.,4.116821`6.},{11.69938`6.,8.262689`6.},{-0.174684`6.,7.725432`6.}},{{11.886212`6.,0},{0,0},{2.511852`6.,-7.307764`6.},{11.886212`6.,0}},{{0,0},{11.886212`6.,0},{11.150507`6.,4.116821`6.},{0,0}},{{13.837044`6.,-3.699153`6.},{2.511852`6.,-7.307764`6.},{7.123757`6.,-13.508017`6.},{13.837044`6.,-3.699153`6.}},{{11.886212`6.,0},{2.511852`6.,-7.307764`6.},{13.837044`6.,-3.699153`6.},{11.886212`6.,0}},{{16.818847`6.,-6.631443`6.},{7.123757`6.,-13.508017`6.},{13.400358`6.,-18.015466`6.},{16.818847`6.,-6.631443`6.}},{{13.837044`6.,-3.699153`6.},{7.123757`6.,-13.508017`6.},{16.818847`6.,-6.631443`6.},{13.837044`6.,-3.699153`6.}},{{20.550144`6.,-8.520067`6.},{13.400358`6.,-18.015466`6.},{20.749153`6.,-20.404613`6.},{20.550144`6.,-8.520067`6.}},{{16.818847`6.,-6.631443`6.},{13.400358`6.,-18.015466`6.},{20.550144`6.,-8.520067`6.},{16.818847`6.,-6.631443`6.}},{{24.678706`6.,-9.186741`6.},{20.749153`6.,-20.404613`6.},{28.476427`6.,-20.449927`6.},{24.678706`6.,-9.186741`6.}},{{20.550144`6.,-8.520067`6.},{20.749153`6.,-20.404613`6.},{24.678706`6.,-9.186741`6.},{20.550144`6.,-8.520067`6.}},{{2.00429`6.,15.139263`6.},{13.481016`6.,12.046239`6.},{6.33123`6.,21.541637`6.},{2.00429`6.,15.139263`6.}},{{24.678706`6.,-9.186741`6.},{28.476427`6.,-20.449927`6.},{28.814803`6.,-8.568532`6.},{24.678706`6.,-9.186741`6.}},{{-0.174684`6.,7.725432`6.},{11.69938`6.,8.262689`6.},{2.00429`6.,15.139263`6.},{-0.174684`6.,7.725432`6.}},{{2.00429`6.,15.139263`6.},{11.69938`6.,8.262689`6.},{13.481016`6.,12.046239`6.},{2.00429`6.,15.139263`6.}}};
m2={{{8.794287`6.,-5.05218`6.},{4.319607`6.,-7.105812`6.},{7.513115`6.,-9.805995`6.},{8.794287`6.,-5.05218`6.}},{{6.586082`6.,-2.735085`6.},{4.319607`6.,-7.105812`6.},{8.794287`6.,-5.05218`6.},{6.586082`6.,-2.735085`6.}},{{11.446238`6.,-6.844457`6.},{7.513115`6.,-9.805995`6.},{11.209435`6.,-11.762189`6.},{11.446238`6.,-6.844457`6.}},{{8.794287`6.,-5.05218`6.},{7.513115`6.,-9.805995`6.},{11.446238`6.,-6.844457`6.},{8.794287`6.,-5.05218`6.}},{{-0.103419`6.,12.352884`6.},{4.819321`6.,12.435316`6.},{1.609089`6.,16.16822`6.},{-0.103419`6.,12.352884`6.}},{{11.446238`6.,-6.844457`6.},{11.209435`6.,-11.762189`6.},{14.419667`6.,-8.029285`6.},{11.446238`6.,-6.844457`6.}},{{-0.961958`6.,8.259916`6.},{3.829704`6.,9.391345`6.},{-0.103419`6.,12.352884`6.},{-0.961958`6.,8.259916`6.}},{{-0.103419`6.,12.352884`6.},{3.829704`6.,9.391345`6.},{4.819321`6.,12.435316`6.},{-0.103419`6.,12.352884`6.}},{{-0.926947`6.,4.07802`6.},{3.512721`6.,6.206283`6.},{-0.961958`6.,8.259916`6.},{-0.926947`6.,4.07802`6.}},{{-0.961958`6.,8.259916`6.},{3.512721`6.,6.206283`6.},{3.829704`6.,9.391345`6.},{-0.961958`6.,8.259916`6.}},{{0,0},{3.882987`6.,3.026975`6.},{-0.926947`6.,4.07802`6.},{0,0}},{{-0.926947`6.,4.07802`6.},{3.882987`6.,3.026975`6.},{3.512721`6.,6.206283`6.},{-0.926947`6.,4.07802`6.}},{{4.92343`6.,0},{0,0},{1.776148`6.,-3.78613`6.},{4.92343`6.,0}},{{0,0},{4.92343`6.,0},{3.882987`6.,3.026975`6.},{0,0}},{{6.586082`6.,-2.735085`6.},{1.776148`6.,-3.78613`6.},{4.319607`6.,-7.105812`6.},{6.586082`6.,-2.735085`6.}},{{4.92343`6.,0},{1.776148`6.,-3.78613`6.},{6.586082`6.,-2.735085`6.},{4.92343`6.,0}}};
m3={{{-1.245331`6.,2.948601`6.},{1.182071`6.,5.444158`6.},{-2.257004`6.,5.985313`6.},{-1.245331`6.,2.948601`6.}},{{-2.257004`6.,5.985313`6.},{1.182071`6.,5.444158`6.},{0.35756`6.,8.284038`6.},{-2.257004`6.,5.985313`6.}},{{0,0},{2.225465`6.,2.677198`6.},{-1.245331`6.,2.948601`6.},{0,0}},{{-1.245331`6.,2.948601`6.},{2.225465`6.,2.677198`6.},{1.182071`6.,5.444158`6.},{-1.245331`6.,2.948601`6.}},{{3.481391`6.,0},{0,0},{1.471409`6.,-2.842544`6.},{3.481391`6.,0}},{{0,0},{3.481391`6.,0},{2.225465`6.,2.677198`6.},{0,0}},{{4.942205`6.,-2.57114`6.},{1.471409`6.,-2.842544`6.},{3.15994`6.,-5.561728`6.},{4.942205`6.,-2.57114`6.}},{{3.481391`6.,0},{1.471409`6.,-2.842544`6.},{4.942205`6.,-2.57114`6.},{3.481391`6.,0}},{{6.599014`6.,-5.020573`6.},{3.15994`6.,-5.561728`6.},{5.055315`6.,-8.141002`6.},{6.599014`6.,-5.020573`6.}},{{4.942205`6.,-2.57114`6.},{3.15994`6.,-5.561728`6.},{6.599014`6.,-5.020573`6.},{4.942205`6.,-2.57114`6.}},{{8.441735`6.,-7.333389`6.},{5.055315`6.,-8.141002`6.},{7.145997`6.,-10.564665`6.},{8.441735`6.,-7.333389`6.}},{{6.599014`6.,-5.020573`6.},{5.055315`6.,-8.141002`6.},{8.441735`6.,-7.333389`6.},{6.599014`6.,-5.020573`6.}},{{-3.02886`6.,9.091651`6.},{-0.243049`6.,11.179554`6.},{-3.556203`6.,12.248708`6.},{-3.02886`6.,9.091651`6.}},{{8.441735`6.,-7.333389`6.},{7.145997`6.,-10.564665`6.},{10.459152`6.,-9.495511`6.},{8.441735`6.,-7.333389`6.}},{{-2.257004`6.,5.985313`6.},{0.35756`6.,8.284038`6.},{-3.02886`6.,9.091651`6.},{-2.257004`6.,5.985313`6.}},{{-3.02886`6.,9.091651`6.},{0.35756`6.,8.284038`6.},{-0.243049`6.,11.179554`6.},{-3.02886`6.,9.091651`6.}}};
hiperboloid5[a_,c_,al_,n_,k_]:=Module[{betha=2Pi/n,stept=2Pi/n,r0=2aCos[al/2]},r[ii_]:=r0/Cos[iibetha/2];zk[ii_]:=If[ii>=0,1,-1]cSqrt[(r[ii]^2/(2a^2)-(1+Cos[al]))/(1-Cos[al])];Flatten[Table[RotateLeft[Table[{((c-(zk[jj]))Cos[tbetha]+(c+zk[jj])Cos[tbetha-al])a/c,((c-(zk[jj]))Sin[tbetha]+(c+(zk[jj]))Sin[tbetha-al])a/c,(zk[jj])}//N,{t,0,n-1}],jj],{jj,-k/2,k/2}],1]]
celica[n_,k_]:=Table[Table[{in+j,in+Mod[j,n]+1,(i+1)n+j},{j,1,n}],{i,0,k}]
celica2[n_,k_]:=Table[Table[{in+Mod[j,n]+1,(i+1)n+Mod[j,n]+1,(i+1)n+j},{j,1,n}],{i,0,k}]
celice[n_,k_]:=Sort[Flatten[Join[celica[n,k],celica2[n,k]],1]]
photo=ImageResize
,400;
Manipulate[With[{hip={celice[n,n/2],N[hiperboloid5[1,2,Pi/2,n,n/2+2]]}},Pane[Switch[nt,2,Graphics[{Map[Line,m1],Translate[Map[Line,m2],{16,4}],Translate[Rotate[Map[Line,m3],-0.3,{0,0}],{24,12}],Translate[Map[Line,m2],{-6,-11}],Translate[Rotate[Map[Line,m3],1,{0,0}],{6,23}]},ImageSize->300],1,Graphics3D[Polygon[Map[hip[[2,#]]&,hip[[1]]]],Boxed->False,SphericalRegion->True,ImageSize->{380,380}],3,photo],ImageSize->{400,400},Alignment->Center]],{{n,8,"division"},Range[8,16,4],Enabled->nt==1},{{nt,1,""},{1->"surface",2->"net for 8 divisions",3->"photo"}},SaveDefinitions->True]
Mesh
Mesh