Displaying Dandelin spheres touching hyperbola boundary patches in cone - Mathematica Stack Exchange most recent 30 from mathematica.stackexchange.com 2019-09-18T06:04:18Z https://mathematica.stackexchange.com/feeds/question/119007 https://creativecommons.org/licenses/by-sa/4.0/rdf https://mathematica.stackexchange.com/q/119007 1 Displaying Dandelin spheres touching hyperbola boundary patches in cone Narasimham https://mathematica.stackexchange.com/users/19067 2016-06-21T17:55:53Z 2018-09-29T14:00:11Z <p>Can we show the <a href="https://math.stackexchange.com/questions/1833973/prove-that-the-directrix-focus-and-focus-focus-definitions-are-equivalent">Dandelin spheres</a> scenario (shown by user <a href="https://math.stackexchange.com/users/332390/hrhm">Hrhm</a> for case of ellipse) for a hyperbola as well? I'm not happy with what we get to see of this on the net. </p> <p>The double nappe/sheet of a cone is cut by a plane inclined at angle <span class="math-container">$\beta &lt; 2 \alpha$</span> ( cutting plane inclination to a generator of cone, cone vertex angle ) cutting both nappes with hyperbolas as intersecting arcs.</p> <p>The Dandelin spheres are placed in each cone tangentially, <a href="http://www.clowder.net/hop/Dandelin/ellhyp.gif" rel="nofollow noreferrer">foci <em>outside</em> directrix planes</a> The cutting plane contacts spheres at foci so that focal distance <em>difference</em> from hyperbola is constant.</p> <p>I can also do this using <code>ContourPlot3D</code>, however you can make it with a great image quality.</p> <p>Proper choice of <span class="math-container">$\alpha, \beta$</span> so that the difference of line segments can be convincingly visualized.</p> <p>EDIT 1:</p> <p>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.</p> <pre><code>{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-&gt; Opacity[.35]]; Plne=ContourPlot3D[ z ==-x Tan[bt]+ H,{x,-PltLim,PltLim},{y,-PltLim,PltLim},{z,-PltLim,PltLim},ContourStyle-&gt; Opacity[.9]]; Show[{S1,S2,Keg,Plne},PlotRange-&gt;All, Boxed-&gt;False,Axes-&gt;None] </code></pre> <p><img src="https://i.stack.imgur.com/KMngg.png" alt="my attempt"></p> https://mathematica.stackexchange.com/questions/119007/-/119029#119029 3 Answer by J. M. will be back soon for Displaying Dandelin spheres touching hyperbola boundary patches in cone J. M. will be back soon https://mathematica.stackexchange.com/users/50 2016-06-21T22:47:27Z 2016-06-21T22:47:27Z <p>Here's a starting point. Figuring out what I exactly did is left as an exercise.</p> <pre><code>With[{r = 5/2, φ = π/4, s1 = 3/4, s2 = 3/2}, Graphics3D[{{Directive[CapForm[None], Opacity[2/3]], Tube[{{0, 0, -r Cot[φ/2]}, {0, 0, 0}, {0, 0, r Cot[φ/2]}}, {r, 0, r}]}, {Sphere[{0, 0, s1 Csc[φ/2]}, s1], Sphere[{0, 0, -s2 Csc[φ/2]}, s2]}, {Directive[EdgeForm[], Opacity[1/2, LightBlue]], InfinitePlane[#1, {#2 - #1, {0, 1, 0}}] &amp;[ {s1/(s1 + s2) Sqrt[(s1 + s2)^2 - (s1 - s2)^2 Sin[φ/2]^2], 0, s1 Csc[φ/2] - s1 (s1 - s2) Sin[φ/2]/(s1 + s2)}, {s2/(s1 + s2) Sqrt[(s1 + s2)^2 - (s1 - s2)^2 Sin[φ/2]^2], 0, s2 (s2 - s1) Sin[φ/2]/(s1 + s2) - s2 Csc[φ/2]}]}}, Boxed -&gt; False]] </code></pre> <p><img src="https://i.stack.imgur.com/nU3bo.png" alt="Dandelin spheres, hyperbolic case"></p> https://mathematica.stackexchange.com/questions/119007/-/119086#119086 1 Answer by stan wagon for Displaying Dandelin spheres touching hyperbola boundary patches in cone stan wagon https://mathematica.stackexchange.com/users/2902 2016-06-22T13:59:29Z 2016-06-22T14:02:31Z <p>Why not go to the WRI demo site and search for demos involving the Dandelin spheres? You can download the source code and see how some others did it.</p> <p>For example: <a href="http://demonstrations.wolfram.com/DandelinSpheresForAnEllipse/" rel="nofollow">http://demonstrations.wolfram.com/DandelinSpheresForAnEllipse/</a></p>