Volume of a torus

[Moontanman]

Not home work but does anyone know a formula for the volume of a torus and the surface area? I'm probably going to need to figure the force on the inside surface of a rotating torus as well so a formula for that would be helpful as well. I am pretty much restricted to algebra and geometry for math skills...

[swansont]

Volume of a simple shape like this is the cross sectional are multiplied by the length. so (pi*r²) * (2*pi*R). r is the radius of the cross section, R is the radius of the torus. Similarly, area is L*w, or (2*pi*r)*(2*pi*R)

 

Just like if it were a rolled-up rectangle. The stretching/squeezing of the outer/inner sides cancel. You can think of R as the average radius, (R₁ +R₂)/2, which would pop out if you solved it rigorously

[Moontanman]

Volume of a simple shape like this is the cross sectional are multiplied by the length. so (pi*r²) * (2*pi*R). r is the radius of the cross section, R is the radius of the torus. Similarly, area is L*w, or (2*pi*r)*(2*pi*R)

 

Just like if it were a rolled-up rectangle. The stretching/squeezing of the outer/inner sides cancel. You can think of R as the average radius, (R₁ +R₂)/2, which would pop out if you solved it rigorously

 

 

Thanks, but it has occurred to me the torus will have to have a pear shaped cross section to be both strong and efficient. Any chance there is a formula for that? I may just approximate by using a large outer torus and a small inner one...

[Sensei]

 

Thanks, but it has occurred to me the torus will have to have a pear shaped cross section to be both strong and efficient. Any chance there is a formula for that? I may just approximate by using a large outer torus and a small inner one...

 

In such cases I am simply making object in 3D application, then using boolean tool or cutting tool, and running area counting tool on newly made cross section polygons.

3D apps also come with volume calculating tools. The more geometry at beginning, the more precise result will be.

 

[Moontanman]

 

In such cases I am simply making object in 3D application, then using boolean tool or cutting tool, and running area counting tool on newly made cross section polygons.

3D apps also come with volume calculating tools. The more geometry at beginning, the more precise result will be.

 

 

 

Is this software available online?

[Sensei]

Is this software available online?

 

It is LightWave 3D

https://www.lightwave3d.com/

It costs $1500.

 

There are other 3D applications: Maya, Cinema 4D, 3D Studio Max, Houdini, etc. etc.

 

I am writing mine own plugins for it, if needed.

They (usually) analyze polygons and points data, and output new geometry, or showing thing that they calculated (like area of polygon(s), or volume of 3d object).

 

Making area counting tool is very easy: just scan all selected polygons, and add together their areas one by one.

Hope so I don't have to ask you whether you know formula of area of triangle..

Then show final result in window.

 

Showed on anim gif above boolean tool is standard tool (Modeler's Construct > Boolean).

Boolean tools regardless of 3d application always have at least modes: unify, subtract, intersect.

Edited June 26, 2014 by Sensei

[Moontanman]

Well the software is way out of my league but I don't need perfection but I would like to reasonably accurate when describing the world I am building.

[Sensei]

If you're working at university you can get educational license for $195.

"LightWave 11.6 Educational Available for Students and Faculty Staff"

Trial has 30 days. Then demo is probably limited by not able to save objects or so. Don't remember exactly.

 

There are other apps like free Blender. But people often complain about its interface.

Edited June 26, 2014 by Sensei

[Moontanman]

If you're working at university you can get educational license for $195.

"LightWave 11.6 Educational Available for Students and Faculty Staff"

Trial has 30 days. Then demo is probably limited by not able to save objects or so. Don't remember exactly.

 

There are other apps like free Blender. But people often complain about its interface.

 

 

I'm not a student, just trying to get some perspective on a story I am writing..