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Two identical right circular cones are resting on their curved surface as shown in the second row of the image above.

Let V be the top vertex and O (O') be the center of base circles and A(A') be the horizontal point on the circumference towards each other sides.

If the vertex of the cones are raised at a horizontal level of the base center, then the top and side view would look like in the first row image above.

Now, the cones are made to touch each other in the resting position of the second row.

It can be observed (experimentally/practically/DIY basis) that when the cones are touching each other such that their vertex as well as the curved surface too touch each other THEN THE BASE CIRCLE WILL TOUCH EACH OTHER at a point B(B')

1) Find the angle BOA (or B'O'A') in terms of the semi vertex angle of the cone?

2) Also find the elevation of point B(or B') over the vertex, there is a maximum value for this elevation, also find that!

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Let a cone be resting on a plane, along generatrix $VB$ (see diagram below), and the perpendicular to the plane at $V$ meet line $BO$ at point $G$. From the similitude of triangles $GVO$ and $OVB$ we get: $$ OG={OV^2\over OB}. $$

Let now $GT$ be a line tangent to the base of the cone: the reflection of the cone about plane $GVT$ is tangent to the given cone at $T$. Hence the requested angle is $\angle AOT$ and: $$ \sin(\angle AOT)=\sin(\angle OGT)={OT\over OG}={OB\over OV^2/OB}= {OB^2\over OV^2}=\tan^2(\angle OVB). $$

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EDIT.

Here's an animation made with GeoGebra, showing that the above formula is in perfect agreement with the angles computed by GeoGebra.

enter image description here

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  • $\begingroup$ This means that the angle in question can not be obtained if semi vertex angle is greater than 45 degree, but this seems impractical! $\endgroup$ Commented Jun 16, 2018 at 21:29
  • $\begingroup$ Two equal cones with semi vertex angle greater than 45° cannot be arranged in the way you want. $\endgroup$ Commented Jun 16, 2018 at 22:02
  • $\begingroup$ I am getting $$y = {\pi\over 2} - {2* \tan^{-1}\ {((1 + \tan^{4}(x))^{0.5} - (\tan(x))^{2}) }}$$ where x = $\angle OVB$ & y = $\angle AOT$, can this expression be checked! $\endgroup$ Commented Jun 17, 2018 at 4:05
  • $\begingroup$ Your formula doesn't give the same result as mine, so it must be wrong: I checked mine with GeoGebra and it is correct. Moreover, it is obvious (just make a sketch) that $y=\pi/2$ for $x=\pi/4$, but your formula gives a different result. As it is obvious that your two cones cannot exist if $x>\pi/4$. I don't understand why you doubt of my formula: if its derivation is not clear I can explain it better. $\endgroup$ Commented Jun 17, 2018 at 9:09

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