Visualizations and graphics for Tritangents

Real space sextic with five ovals

colors = ColorData[97,"ColorList"][[1;;5]];
s2 := x^2 + y^2 + z^2-25
c3 := (x + Sqrt[3]) (Sqrt[3] y - x + 3) (Sqrt[3] y + x - 3) + 2
boxbnds = 5;
curve = ContourPlot3D[{c3 == 0, s2 == 0}, {x, -boxbnds, boxbnds}, {y, -boxbnds, boxbnds}, {z, -boxbnds, boxbnds},
  ContourStyle -> {None, Directive[Specularity[0],RGBColor[1,1,0.95,0.7],Glow[Blue]]}, Mesh -> None, 
 PlotPoints -> 24, 
 BoundaryStyle -> {1 -> None, 2 -> None, {1, 2} -> {Red, Thick}}, 
 Boxed -> False, Axes -> False] ;
Trit[a_,b_,c_] = ContourPlot3D[{a*x+b*y+c*z+1==0},{x,-boxbnds,boxbnds},{y,-boxbnds,boxbnds},{z,-boxbnds,boxbnds},
ContourStyle->{},Mesh->None];
isTotallyReal[abc_] := 
Module[{plane,TP},
plane = abc[[1]]*x+abc[[2]]*y+abc[[3]]*z+1;
TP=Values[NSolve[{s2==0,c3==0,plane==0},{x,y,z}]];
TP=DeleteDuplicates[TP,(Norm[#1-#2]<0.01)&];
Norm[Im[TP[[1]]]]<0.01 && Norm[Im[TP[[2]]]]<0.01 && Norm[Im[TP[[3]]]]<0.01]

ShowTritangentPlane[a_,b_,c_] = Show[curve,Trit[a,b,c],ViewPoint->500*Cross[{1,1,1},{a,b,c}]]&
(*ShowTritangentPlane[a_,b_,c_] = Show[curve,Trit[a,b,c]]&*)

Show[curve]

tritangents={{-0.3365059720,-0.6182905947,0},{-0.3701289525,-0.6017252075,0},{-0.3255116642,-0.5667926316,0},{-0.3274456091,-0.5633494466,0},{-0.09413659936,-0.5169813351,0},{-0.1204422416,-0.3937642102,0},{-0.4189756212,-0.3735043455,0},{-0.2891509500,-0.3240808109,0},{0.4673656191,-0.1512498153,0},{0.3859189627,-0.07802271630,0},{0.6055490851,-0.01652654561,0},{0.5662681083,-0.001572494927,0},{0.5662681083,0.001572494927,0},{0.6055490851,0.01652654561,0},{0.3859189627,0.07802271630,0},{0.4673656191,0.1512498153,0},{-0.2891509500,0.3240808109,0},{-0.4189756212,0.3735043455,0},{-0.1204422416,0.3937642102,0},{-0.09413659936,0.5169813351,0},{-0.3274456091,0.5633494466,0},{-0.3255116642,0.5667926316,0},{-0.3701289525,0.6017252075,0},{-0.3365059720,0.6182905947,0},{5.473378457,0,-3.053343695},{-2.766177684,0,-1.204225559},{-1.418572304,0,-1.192601621},{-0.7460757647,0,-0.4901223544},{0.01413809166,0,-0.3039850362},{0.07582117476,0,-0.2643009922},{-0.04177070926,0,-0.2455835856},{0.008104581289,0,-0.2293102620},{0.006803689025,0,-0.2093378481},{0.2555878981,0,-0.1230098123},{0.5712096968,0,-0.01348915565},{0.5992135284,0,-0.003771835639},{0.5992135284,0,0.003771835639},{0.5712096968,0,0.01348915565},{0.2555878981,0,0.1230098123},{0.006803689025,0,0.2093378481},{0.008104581289,0,0.2293102620},{-0.04177070926,0,0.2455835856},{0.07582117476,0,0.2643009922},{0.01413809166,0,0.3039850362},{-0.7460757647,0,0.4901223544},{-1.418572304,0,1.192601621},{-2.766177684,0,1.204225559},{5.473378457,0,3.053343695},{4.203608831,-6.908335439,-4.512525297},{4.203608831,6.908335439,-4.512525297},{-5.164861587,-0.9518251070,-2.447151422},{-5.164861587,0.9518251070,-2.447151422},{0.7840035946,-2.061793704,-1.034367894},{0.7840035946,2.061793704,-1.034367894},{1.341282297,-1.530900059,-0.9523200748},{1.341282297,1.530900059,-0.9523200748},{0.4795734808,-0.7672646473,-0.7761907043},{0.4795734808,0.7672646473,-0.7761907043},{0.7990575843,-1.343383850,-0.6847864208},{0.7990575843,1.343383850,-0.6847864208},{-0.8822713242,-0.2131469656,-0.6314932674},{-0.8822713242,0.2131469656,-0.6314932674},{0.2153229466,-0.6579514566,-0.4957155524},{0.2153229466,0.6579514566,-0.4957155524},{0.4775739269,-0.4794923228,-0.4801749384},{0.4775739269,0.4794923228,-0.4801749384},{0.3052153533,-0.5048126571,-0.3970628762},{0.3052153533,0.5048126571,-0.3970628762},{-0.02090243152,-0.05774095951,-0.2670030734},{-0.02090243152,0.05774095951,-0.2670030734},{0.03514904761,-0.04598676573,-0.2436302652},{0.03514904761,0.04598676573,-0.2436302652},{-0.1209849477,-0.2249353359,-0.1333431969},{-0.1209849477,0.2249353359,-0.1333431969},{-0.1095575965,-0.4294043599,-0.04656500458},{-0.1095575965,0.4294043599,-0.04656500458},{-0.3294783552,-0.3365633511,-0.04499806683},{-0.3294783552,0.3365633511,-0.04499806683},{0.4084305850,-0.1017737448,-0.04176670329},{0.4084305850,0.1017737448,-0.04176670329},{-0.3307194220,-0.5707897158,-0.01498601766},{-0.3307194220,0.5707897158,-0.01498601766},{-0.3480285820,-0.6026469542,-0.004336703677},{-0.3480285820,0.6026469542,-0.004336703677},{-0.3480285820,-0.6026469542,0.004336703677},{-0.3480285820,0.6026469542,0.004336703677},{-0.3307194220,-0.5707897158,0.01498601766},{-0.3307194220,0.5707897158,0.01498601766},{0.4084305850,-0.1017737448,0.04176670329},{0.4084305850,0.1017737448,0.04176670329},{-0.3294783552,-0.3365633511,0.04499806683},{-0.3294783552,0.3365633511,0.04499806683},{-0.1095575965,-0.4294043599,0.04656500458},{-0.1095575965,0.4294043599,0.04656500458},{-0.1209849477,-0.2249353359,0.1333431969},{-0.1209849477,0.2249353359,0.1333431969},{0.03514904761,-0.04598676573,0.2436302652},{0.03514904761,0.04598676573,0.2436302652},{-0.02090243152,-0.05774095951,0.2670030734},{-0.02090243152,0.05774095951,0.2670030734},{0.3052153533,-0.5048126571,0.3970628762},{0.3052153533,0.5048126571,0.3970628762},{0.4775739269,-0.4794923228,0.4801749384},{0.4775739269,0.4794923228,0.4801749384},{0.2153229466,-0.6579514566,0.4957155524},{0.2153229466,0.6579514566,0.4957155524},{-0.8822713242,-0.2131469656,0.6314932674},{-0.8822713242,0.2131469656,0.6314932674},{0.7990575843,-1.343383850,0.6847864208},{0.7990575843,1.343383850,0.6847864208},{0.4795734808,-0.7672646473,0.7761907043},{0.4795734808,0.7672646473,0.7761907043},{1.341282297,-1.530900059,0.9523200748},{1.341282297,1.530900059,0.9523200748},{0.7840035946,-2.061793704,1.034367894},{0.7840035946,2.061793704,1.034367894},{-5.164861587,-0.9518251070,2.447151422},{-5.164861587,0.9518251070,2.447151422},{4.203608831,-6.908335439,4.512525297},{4.203608831,6.908335439,4.512525297}};
totallyRealTritangents = Select[tritangents,isTotallyReal];
Length[totallyRealTritangents]
GraphicsGrid[Partition[(ShowTritangentPlane@@#)[]&/@totallyRealTritangents[[1;;30]],UpTo[5]],ImageSize->Full]
GraphicsGrid[Partition[(ShowTritangentPlane@@#)[]&/@totallyRealTritangents[[31;;60]],UpTo[5]],ImageSize->Full]
GraphicsGrid[Partition[(ShowTritangentPlane@@#)[]&/@totallyRealTritangents[[61;;90]],UpTo[5]],ImageSize->Full]
GraphicsGrid[Partition[(ShowTritangentPlane@@#)[]&/@totallyRealTritangents[[91;;108]],UpTo[5]],ImageSize->Full]

discardedTritangents = Complement[tritangents,totallyRealTritangents];
Length[discardedTritangents]
GraphicsGrid[Partition[(ShowTritangentPlane@@#)[]&/@totallyRealTritangents[[91;;108]],UpTo[5]],ImageSize->Full]