DandelineSpheres-Parabola::usage="Sphere tangent to a plane parallel to the generatrix and a cone, its point of contact with the plane determines the focus of a parabola in space formed by the intersection of the plane and the cone. The distances
PF
and
Pl
(point to directrix) must be equal. Reference: Personal. ";
In[]:=
​​Manipulate​​l2[t_]:={-3,t,1};l21[t_]:=-3,2
2
Csc
t
2

Sin[t],1;​​k[x_,y_]:=1;​​j[x_,y_]:=x+4;​​γ[t_]:=2Cos[t]
2
Csc
t
2

,2
2
Csc
t
2

Sin[t],4+2Cos[t]
2
Csc
t
2

;ζ[t_]:=-
4-4
2
t
2
,t,1,
4-4
2
t
2
,t,1;​​QP={-3,0,1}-γ[θ1];PC=Cross[QP,{0,1,0}];N1=Norm[PC];N2=Norm[{0,1,0}];​​d4[t_]:=N[Norm[{-1,0,3}-γ[θ1]]];​​d5[t_]:=N
N1
N2
;​​p=Grid[
PF
,
Pl
,{d4[θ1],d5[θ1]},FrameAll,ItemSize->15,ItemStyle->15];​​​​Show[surf1=ParametricPlot3D[{vCos[u],vSin[u],v},{u,0,2π},{v,-20,20},PlotStyleDirective[Green,Specularity[White,20],Opacity[0.1]],MeshFalse,PerformanceGoal"Quality"],​​surf3=ParametricPlot3D[
2
Cos[u]Sin[v],
2
Sin[u]Sin[v],
2
Cos[v]+2,{u,0,2π},{v,0,π},ColorFunction"LightTemperatureMap",MeshFalse,PerformanceGoal"Quality"],​​surf4=Plot3D[j[x,y],{x,-4,4},{y,-6,6},PlotStyleDirective[Yellow,Specularity[White,20],Opacity[0.2]],MeshFalse,PerformanceGoal"Quality"],surf5=Plot3D[k[x,y],{x,-4,4},{y,-6,6},PlotStyleDirective[Red,Specularity[White,20],Opacity[0.2]],MeshFalse,PerformanceGoal"Quality"],curv1=ParametricPlot3D[γ[t],{t,0,θ1},ColorFunctionFunction[{x,y,z},Hue[z]],PerformanceGoal"Quality"],curv2=ParametricPlot3D[ζ[t],{t,-1,1},PerformanceGoal"Quality"],Graphics3D[Tube[{l2[-10],l2[10]},0.07]],​​Graphics3D[{Black,Ball[{-1,0,3},0.12]}],Graphics3D[{Red,Ball[N[γ[θ1]],0.12]}],Graphics3D[{Black,Ball[N[l21[θ1]],0.12]}],​​Graphics3D[{Black,Dashed,Line[{N[γ[θ1]],{-1,0,3}}]}],Graphics3D[{Black,Dashed,Line[{N[γ[θ1]],l21[θ1]}]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},MeshFalse,AxesTrue,BoxedFalse,PlotRange4,BoxRatiosAutomatic,ViewPoint{-3,2,1},ImageSizeFull],Style["Dandeline Spheres-Parabola:",Bold,Medium],{θ1,0.6,"Angle (θ)"},-
3
2
π,
3
2
π,0.1,Delimiter,{{p,2,"Distance from point to focus and from point to directrix"}},ControlPlacementTop​​​​​​​​
Out[]=
​
Dandeline Spheres-Parabola:
Angle (θ)
Distance from point to focus and from point to directrix
PF
Pl
3.61654
3.61654