Geometry problem: a cone meeting a cylindre.

[Vilestag]

Hi,

I have a cone on the z axis with his summit on height h meeting a cylinder on the x axis. The expressions should be:

cylinder: y²+z²=r²

cone: x²+y² =(z-h)²tan(phi)²

If we consider any straight line on the cone, what is the length of this line inside the cylinder?

Is it possible to get a theoretical exprssion of this?

I tried the approche of the distance between two points, but i need a third equation to know the three coordinates of each points. Any ideas?

Thanks a lot,

Alx

 

[tiny-tim]

Welcome to PF!

Hi Alx! Welcome to PF!

(have a phi: φ )

cylinder: y²+z²=r²

cone: x²+y² =(z-h)²tan(phi)²
…
… i need a third equation to know the three coordinates of each points

the third equation will be a linear one, for a particular line

 

[Vilestag]

Thanks for your response.

In fact, I need an expression of the length for ANY line if possible at all.

My problem goes much deeper: I need to find the mean length for all lines for all Φ...

All I have to do is to find an expression only in function of Φ and integrate it on all Φ. As simple as it sounds, I can't figure it out, because it's far from simple. I've done it in 2D (a triangle passing through a circle) and it worked, so i know my approach is good.

Anyway, I'll take any hint I get.

Tanks again,

Alx

 

[tiny-tim]

Hi Alx!

I know it's complicated, but you'll just have to work through it.

Use y = xtanθ, find the length, and average over θ

 

[zgozvrm]

cone: x²+y² =(z-h)²tan(phi)²

I think you mean ...

$$x^2 + y^2 = \frac{(h-z)^2}{tan^2\phi}$$