Minimum Radius Cylinder for Tangent Line-Contact with Cone

[gwiz]

If I have a 45° right-angle cone and I place it on a table on the conical surface (not the base), there should be a line-contact along the cone (the table is tangent to the conical surface). The table can be seen as a cylinder with an infinite radius, so, my question is, what is the minimum radius cylinder that the cone can lie in, while maintaining the line-contact (remaining tangent)?

 

[zgozvrm]

Does 45° refer to the angle the sides of the cone make with the base, or the maximum angle formed by the "nose" of the cone?

 

[gsal]

I don't understand.

I mean, when you place a cylinder on its side on top of a table, it also has line contact, no matter what the diameter of the cylinder is...to that end, a cone and a cylinder could have line contact, no matter what the radius of either of them.

did I misunderstood the problem? or what?

 

[zgozvrm]

I believe what he is asking is: if you were able to roll up the surface of the table to form a cylinder around the cone without disturbing the way the cone is lying on the table and without intersecting any part of the cone, what is the minimum radius of such a cylinder?

 

[zgozvrm]

So, the problem then becomes...

Find the dimensions of the ellipse formed when viewing the "tilted" cone from the surface of the table. Then, find the dimensions of a circle that will circumscribe the ellipse, intersecting at only one point: the "bottom" of the ellipse, which lies at one end of the minor axis.

 

[gsal]

Oh, I see...after reading the posting I had forgotten about the title...the wrapping of the table onto a cylinder enclosing the cone!...I wrapped the table in the other direction so my cone was left outside the cylinder.

I get it now.

 

[gwiz]

Yes, that is exactly right. I am wrapping the table around the cone, and trying to find the minimum radius that remains tangent to the line contact of the cone. Does anyone know of any existing equations for that, or do I just need to drudge through it, and derive it?