StraightConoid::usage="Find the equation of the OZ axis straight conoid, which has as its directrix the intersection of the surface + + = 1 with the plane + = 1. References:UsunárizBalanzategui,UbaldoandUsunárizSala,Ignacio. ProblemasdeGeometríaAnalíticayDiferencial.p.279;Personal.";
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Development::
Straight conoid equation::
By the definition of a conoid, the generatrix is a line perpendicular to the axis of the conoid:
z==λ;y==mx;{x,mx,λ};Eliminate++==1,+==1,x
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-12m+4+12m+90;-12+4+12+9==0;-12xy+4+12xy+9==0;
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Directrix:
Necesary for the animation:
In[]:=
ζ[z_]:=1-[z_]:=1+>=0,z]
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,z;2
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,z;Reduce[1-22
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-≤z≤;
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Animation:
In[]:=
SwatchLegend[{Orange,Yellow},{"Generatrix","Directrix"}]Manipulatef[x_,y_,z_]:=-12xy+4+12xy+9;ζ[z_]:=1-[z_]:=1+,,PlotStyleYellow/.Line[pts_,rest___]:>Tube[pts,0.05,rest],ParametricPlot3D[z],z,-,,PlotStyleYellow/.Line[pts_,rest___]:>Tube[pts,0.05,rest],ContourPlot3D[f[x,y,z]0,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle->Directive[Green,Opacity[0.2],Specularity[White,30]],MeshFalse,PerformanceGoal->"Quality",PlotPoints60],Graphics3D[{Orange,Thick,InfiniteLine[{{0,0,t1},ζ[t1]}]}],Graphics3D[{Orange,Thick,InfiniteLine[{{0,0,t1},[t1]}]}],Graphics3D[{Black,Ball[{0,0,t1},0.08]}],Graphics3D[{Yellow,Ball[ζ[t1],0.08]}],Graphics3D[{Yellow,Ball[[t1],0.08]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange5,ImageSize900,ViewPoint{15,-3,10},Style["The straight conoid:",Bold,Large],t1,-+0.00002,"Value (t1)",-+0.00001,,0.05,ControlPlacementTop
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,z;ShowParametricPlot3Dζ[z],z,-2
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