10 added 69 characters in body edited Sep 29 '18 at 14:00 J. M. will be back soon♦ 101k1010 gold badges317317 silver badges477477 bronze badges Can we show the Dandelin spheres scenario (shown by Hrhmuser Hrhm for case of ellipse) for a hyperbola as well? I'm not happy with what we get to see of this on the net. The double nappe/sheet of a cone is cut by a plane inclined at angle $$\beta < 2 \alpha$$ ( cutting plane inclination to a generator of cone, cone vertex angle ) cutting both nappes with hyperbolas as intersecting arcs. The Dandelin spheres are placed in each cone tangentially, foci outside directrix planes The cutting plane contacts spheres at foci so that focal distance difference from hyperbola is constant. I can also do this using ContourPlot3D, however you can make it with a great image quality. Proper choice of $$\alpha, \beta$$ so that the difference of line segments can be convincingly visualized. EDIT 1: Some changes from the following need to be made to see points of tangential contact and radius vectors in the plane at an arbitrary point on hyperbola. {s1=5,s2=3,H=-1,al=.8,bt=-2,Z=11.5,PltLim=5.1}; snal=(s1+s2)/Z;al=ArcSin[snal];tnal=Tan[al]; S1=ContourPlot3D[x^2+y^2+(z+s1/snal)^2==s1^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; S2=ContourPlot3D[x^2+y^2+(z-s2/snal)^2==s2^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; Keg=ContourPlot3D[x^2+y^2 -z^2 tnal^2==0,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.35]]; Plne=ContourPlot3D[ z ==-x Tan[bt]+ H,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.9]]; Show[{S1,S2,Keg,Plne},PlotRange->All, Boxed->False,Axes->None]  Can we show the Dandelin spheres scenario (shown by Hrhm for case of ellipse) for a hyperbola as well? I'm not happy with what we get to see of this on the net. The double nappe/sheet of a cone is cut by a plane inclined at angle $$\beta < 2 \alpha$$ ( cutting plane inclination to a generator of cone, cone vertex angle ) cutting both nappes with hyperbolas as intersecting arcs. The Dandelin spheres are placed in each cone tangentially, foci outside directrix planes The cutting plane contacts spheres at foci so that focal distance difference from hyperbola is constant. I can also do this using ContourPlot3D, however you can make it with a great image quality. Proper choice of $$\alpha, \beta$$ so that the difference of line segments can be convincingly visualized. EDIT 1: Some changes from the following need to be made to see points of tangential contact and radius vectors in the plane at an arbitrary point on hyperbola. {s1=5,s2=3,H=-1,al=.8,bt=-2,Z=11.5,PltLim=5.1}; snal=(s1+s2)/Z;al=ArcSin[snal];tnal=Tan[al]; S1=ContourPlot3D[x^2+y^2+(z+s1/snal)^2==s1^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; S2=ContourPlot3D[x^2+y^2+(z-s2/snal)^2==s2^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; Keg=ContourPlot3D[x^2+y^2 -z^2 tnal^2==0,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.35]]; Plne=ContourPlot3D[ z ==-x Tan[bt]+ H,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.9]]; Show[{S1,S2,Keg,Plne},PlotRange->All, Boxed->False,Axes->None]  Can we show the Dandelin spheres scenario (shown by user Hrhm for case of ellipse) for a hyperbola as well? I'm not happy with what we get to see of this on the net. The double nappe/sheet of a cone is cut by a plane inclined at angle $$\beta < 2 \alpha$$ ( cutting plane inclination to a generator of cone, cone vertex angle ) cutting both nappes with hyperbolas as intersecting arcs. The Dandelin spheres are placed in each cone tangentially, foci outside directrix planes The cutting plane contacts spheres at foci so that focal distance difference from hyperbola is constant. I can also do this using ContourPlot3D, however you can make it with a great image quality. Proper choice of $$\alpha, \beta$$ so that the difference of line segments can be convincingly visualized. EDIT 1: Some changes from the following need to be made to see points of tangential contact and radius vectors in the plane at an arbitrary point on hyperbola. {s1=5,s2=3,H=-1,al=.8,bt=-2,Z=11.5,PltLim=5.1}; snal=(s1+s2)/Z;al=ArcSin[snal];tnal=Tan[al]; S1=ContourPlot3D[x^2+y^2+(z+s1/snal)^2==s1^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; S2=ContourPlot3D[x^2+y^2+(z-s2/snal)^2==s2^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; Keg=ContourPlot3D[x^2+y^2 -z^2 tnal^2==0,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.35]]; Plne=ContourPlot3D[ z ==-x Tan[bt]+ H,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.9]]; Show[{S1,S2,Keg,Plne},PlotRange->All, Boxed->False,Axes->None]  9 replaced http://math.stackexchange.com/ with https://math.stackexchange.com/ edited Apr 13 '17 at 12:19 Can we show the Dandelin spheresDandelin spheres scenario (shown by Hrhm for case of ellipse) for a hyperbola as well? I'm not happy with what we get to see of this on the net. The double nappe/sheet of a cone is cut by a plane inclined at angle $$\beta < 2 \alpha$$ ( cutting plane inclination to a generator of cone, cone vertex angle ) cutting both nappes with hyperbolas as intersecting arcs. The Dandelin spheres are placed in each cone tangentially, foci outside directrix planes The cutting plane contacts spheres at foci so that focal distance difference from hyperbola is constant. I can also do this using ContourPlot3D, however you can make it with a great image quality. Proper choice of $$\alpha, \beta$$ so that the difference of line segments can be convincingly visualized. EDIT 1: Some changes from the following need to be made to see points of tangential contact and radius vectors in the plane at an arbitrary point on hyperbola. {s1=5,s2=3,H=-1,al=.8,bt=-2,Z=11.5,PltLim=5.1}; snal=(s1+s2)/Z;al=ArcSin[snal];tnal=Tan[al]; S1=ContourPlot3D[x^2+y^2+(z+s1/snal)^2==s1^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; S2=ContourPlot3D[x^2+y^2+(z-s2/snal)^2==s2^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; Keg=ContourPlot3D[x^2+y^2 -z^2 tnal^2==0,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.35]]; Plne=ContourPlot3D[ z ==-x Tan[bt]+ H,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.9]]; Show[{S1,S2,Keg,Plne},PlotRange->All, Boxed->False,Axes->None]  Can we show the Dandelin spheres scenario (shown by Hrhm for case of ellipse) for a hyperbola as well? I'm not happy with what we get to see of this on the net. The double nappe/sheet of a cone is cut by a plane inclined at angle $$\beta < 2 \alpha$$ ( cutting plane inclination to a generator of cone, cone vertex angle ) cutting both nappes with hyperbolas as intersecting arcs. The Dandelin spheres are placed in each cone tangentially, foci outside directrix planes The cutting plane contacts spheres at foci so that focal distance difference from hyperbola is constant. I can also do this using ContourPlot3D, however you can make it with a great image quality. Proper choice of $$\alpha, \beta$$ so that the difference of line segments can be convincingly visualized. EDIT 1: Some changes from the following need to be made to see points of tangential contact and radius vectors in the plane at an arbitrary point on hyperbola. {s1=5,s2=3,H=-1,al=.8,bt=-2,Z=11.5,PltLim=5.1}; snal=(s1+s2)/Z;al=ArcSin[snal];tnal=Tan[al]; S1=ContourPlot3D[x^2+y^2+(z+s1/snal)^2==s1^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; S2=ContourPlot3D[x^2+y^2+(z-s2/snal)^2==s2^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; Keg=ContourPlot3D[x^2+y^2 -z^2 tnal^2==0,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.35]]; Plne=ContourPlot3D[ z ==-x Tan[bt]+ H,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.9]]; Show[{S1,S2,Keg,Plne},PlotRange->All, Boxed->False,Axes->None]  Can we show the Dandelin spheres scenario (shown by Hrhm for case of ellipse) for a hyperbola as well? I'm not happy with what we get to see of this on the net. The double nappe/sheet of a cone is cut by a plane inclined at angle $$\beta < 2 \alpha$$ ( cutting plane inclination to a generator of cone, cone vertex angle ) cutting both nappes with hyperbolas as intersecting arcs. The Dandelin spheres are placed in each cone tangentially, foci outside directrix planes The cutting plane contacts spheres at foci so that focal distance difference from hyperbola is constant. I can also do this using ContourPlot3D, however you can make it with a great image quality. Proper choice of $$\alpha, \beta$$ so that the difference of line segments can be convincingly visualized. EDIT 1: Some changes from the following need to be made to see points of tangential contact and radius vectors in the plane at an arbitrary point on hyperbola. {s1=5,s2=3,H=-1,al=.8,bt=-2,Z=11.5,PltLim=5.1}; snal=(s1+s2)/Z;al=ArcSin[snal];tnal=Tan[al]; S1=ContourPlot3D[x^2+y^2+(z+s1/snal)^2==s1^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; S2=ContourPlot3D[x^2+y^2+(z-s2/snal)^2==s2^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; Keg=ContourPlot3D[x^2+y^2 -z^2 tnal^2==0,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.35]]; Plne=ContourPlot3D[ z ==-x Tan[bt]+ H,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.9]]; Show[{S1,S2,Keg,Plne},PlotRange->All, Boxed->False,Axes->None]  8 added 53 characters in body edited Jun 23 '16 at 3:11 J. M. will be back soon♦ 101k1010 gold badges317317 silver badges477477 bronze badges Can we show the Dandelin spheres scenario (shown by Hrhm for case of ellipse) for a hyperbola as well? I'm not happy with what we get to see of this on the net. The double nappe/sheet of a cone is cut by a plane inclined at angle $$\beta < 2 \alpha$$ ( cutting plane inclination to a generator of cone, cone vertex angle ) cutting both nappes with hyperbolas as intersecting arcs. The Dandelin spheres are placed in each cone tangentially, foci outside directrix planes The cutting plane contacts spheres at foci so that focal distance difference from hyperbola is constant. I can also do this using ContourPlot3D, however you can make it with a great image quality. Proper choice of $$\alpha, \beta$$ so that the difference of line segments can be convincingly visualized. EDIT 1: Some changes from the following need to be made to see points of tangential contact and radius vectors in the plane at an arbitrary point on hyperbola. {s1=5,s2=3,H=-1,al=.8,bt=-2,Z=11.5,PltLim=5.1}; snal=(s1+s2)/Z;al=ArcSin[snal];tnal=Tan[al]; S1=ContourPlot3D[x^2+y^2+(z+s1/snal)^2==s1^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; S2=ContourPlot3D[x^2+y^2+(z-s2/snal)^2==s2^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; Keg=ContourPlot3D[x^2+y^2 -z^2 tnal^2==0,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.35]]; Plne=ContourPlot3D[ z ==-x Tan[bt]+ H,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.9]]; Show[{S1,S2,Keg,Plne},PlotRange->All, Boxed->False,Axes->None]  Can we show the Dandelin spheres scenario (shown by Hrhm for case of ellipse) for a hyperbola as well? I'm not happy with what we get to see of this on the net. The double nappe/sheet of a cone is cut by a plane inclined at angle $$\beta < 2 \alpha$$ ( cutting plane inclination to a generator of cone, cone vertex angle ) cutting both nappes with hyperbolas as intersecting arcs. The Dandelin spheres are placed in each cone tangentially, foci outside directrix planes The cutting plane contacts spheres at foci so that focal distance difference from hyperbola is constant. I can also do this using ContourPlot3D, however you can make it with a great image quality. Proper choice of $$\alpha, \beta$$ so that the difference of line segments can be convincingly visualized. EDIT 1: Some changes from the following need to be made to see points of tangential contact and radius vectors in the plane at an arbitrary point on hyperbola. {s1=5,s2=3,H=-1,al=.8,bt=-2,Z=11.5,PltLim=5.1}; snal=(s1+s2)/Z;al=ArcSin[snal];tnal=Tan[al]; S1=ContourPlot3D[x^2+y^2+(z+s1/snal)^2==s1^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; S2=ContourPlot3D[x^2+y^2+(z-s2/snal)^2==s2^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; Keg=ContourPlot3D[x^2+y^2 -z^2 tnal^2==0,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.35]]; Plne=ContourPlot3D[ z ==-x Tan[bt]+ H,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.9]]; Show[{S1,S2,Keg,Plne},PlotRange->All, Boxed->False,Axes->None]  Can we show the Dandelin spheres scenario (shown by Hrhm for case of ellipse) for a hyperbola as well? I'm not happy with what we get to see of this on the net. The double nappe/sheet of a cone is cut by a plane inclined at angle $$\beta < 2 \alpha$$ ( cutting plane inclination to a generator of cone, cone vertex angle ) cutting both nappes with hyperbolas as intersecting arcs. The Dandelin spheres are placed in each cone tangentially, foci outside directrix planes The cutting plane contacts spheres at foci so that focal distance difference from hyperbola is constant. I can also do this using ContourPlot3D, however you can make it with a great image quality. Proper choice of $$\alpha, \beta$$ so that the difference of line segments can be convincingly visualized. EDIT 1: Some changes from the following need to be made to see points of tangential contact and radius vectors in the plane at an arbitrary point on hyperbola. {s1=5,s2=3,H=-1,al=.8,bt=-2,Z=11.5,PltLim=5.1}; snal=(s1+s2)/Z;al=ArcSin[snal];tnal=Tan[al]; S1=ContourPlot3D[x^2+y^2+(z+s1/snal)^2==s1^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; S2=ContourPlot3D[x^2+y^2+(z-s2/snal)^2==s2^2,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim}]; Keg=ContourPlot3D[x^2+y^2 -z^2 tnal^2==0,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.35]]; Plne=ContourPlot3D[ z ==-x Tan[bt]+ H,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-> Opacity[.9]]; Show[{S1,S2,Keg,Plne},PlotRange->All, Boxed->False,Axes->None]  7 deleted 5 characters in body edited Jun 22 '16 at 17:20 Narasimham 1,21599 silver badges1717 bronze badges 6 Code for the 4 assembled objects included edited Jun 22 '16 at 16:55 Narasimham 1,21599 silver badges1717 bronze badges 5 deleted 394 characters in body; edited tags edited Jun 22 '16 at 6:29 J. M. will be back soon♦ 101k1010 gold badges317317 silver badges477477 bronze badges 4 added 491 characters in body edited Jun 21 '16 at 18:47 Narasimham 1,21599 silver badges1717 bronze badges 3 deleted 24 characters in body; edited tags edited Jun 21 '16 at 18:05 Narasimham 1,21599 silver badges1717 bronze badges 2 edited tags | link edited Jun 21 '16 at 18:02 J. M. will be back soon♦ 101k1010 gold badges317317 silver badges477477 bronze badges 1 asked Jun 21 '16 at 17:55 Narasimham 1,21599 silver badges1717 bronze badges