Using Tetrahedrons to find the volume of a geodesic dome.

[John Hurt]

Friends,

I have written several online geodesic dome calculators using PI to calculate the dome volume.  Recently I added a geodesic dome volume calculation using tetrahedrons, with 3 edges of the tet being the radius, and the other 3 being the edges of the triangular face.  I am using this equation from Casio for my tet volume calculations:

https://keisan.casio.com/exec/system/1329962711

Everything seems correct for the class 1 2v, 4v, and 6v geodesic domes.  But the 3v and 5v have a different problem.

First, the center of the geodesic sphere is below the floor of a 3v 3/8 dome, and above the floor of a 3v 5/8 dome.  This means that some of the tets are truncated by the dome floor of the 3v 3/8, and there is additional volume between the floor of the 3v 5/8 and center of the 5/8 sphere.  This same problem is also true for the 5v 7/15 and 8/15 dome.

I am not certain that my solution for this problem is correct, but here it is:  My solution is to calculate the volume of the entire 3v geodesic sphere using tetrahedrons, and then use the total tet volume and the height of the 3v 5/8 or 3/8 dome to find the tet dome volume as a truncated sphere using the spherical cap formula:

https://en.wikipedia.org/wiki/Spherical_cap

https://www.fxsolver.com/browse/formulas/Volume+of+Spherical+Dome+%28or+cap%29

Is there a better way to do this?

I am also not certain of the height of a 3v 3/8 or 5/8 dome as compared to the radius, as it has an uneven base.   Do you take the average height of the uneven base?  What is the height of a 3v or 5v dome vs the radius?  The only constants I have are from other online calculators, that were written by people who are probably as much "in the dark" as I am. 

For the 2v, 4v, 6v, the height is the radius, and is expressed with some precision, but the 3v and 5v height constants have much less precision.  If you have precise constants for the heights of these domes, that would be appreciated.

Thanks,

John Hurt

[Dick Fischbeck]

John- Why not stick with a faceted surface? I mean, a geodesic dome is faceted after all. So basically, what you are calling a spherical cap is just a face.

--
--
You received this message because you are subscribed to the "Geodesic Help" Google Group
--
To unsubscribe from this group, send email to GeodesicHelp...@googlegroups.com
--
To post to this group, send email to geodes...@googlegroups.com
--
For more options, visit http://groups.google.com/group/geodesichelp?hl=en

---
You received this message because you are subscribed to the Google Groups "Geodesic Help Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email to geodesichelp...@googlegroups.com.
To view this discussion on the web visit https://groups.google.com/d/msgid/geodesichelp/645ab199-b892-4c0b-b239-7239178b2049n%40googlegroups.com.

[Dick Fischbeck]

One of my top ten geometric images or concepts is "Radiant Photons."

http://www.rwgrayprojects.com/synergetics/plates/figs/plate10.html

[Adrian Rossiter]

Hi John

On Mon, 3 Apr 2023, John Hurt wrote:
> calculate the dome volume. Recently I added a geodesic dome volume
> calculation using tetrahedrons, with 3 edges of the tet being the radius,
> and the other 3 being the edges of the triangular face. I am using this
> equation from Casio for my tet volume calculations:
>
> https://keisan.casio.com/exec/system/1329962711
>
> Everything seems correct for the class 1 2v, 4v, and 6v geodesic domes.
> But the 3v and 5v have a different problem.
>
> First, the center of the geodesic sphere is below the floor of a 3v 3/8
> dome, and above the floor of a 3v 5/8 dome. This means that some of the
> tets are truncated by the dome floor of the 3v 3/8, and there is additional
> volume between the floor of the 3v 5/8 and center of the 5/8 sphere. This
> same problem is also true for the 5v 7/15 and 8/15 dome.

You can calculate the volume inside a closed surface by dividing the
surface into triangles and joing them to a point O and calculating the
sum of the *signed* volumes of the tetrahdera formed.

Essentially, paint the inside of the dome (including its floor) white,
and the outside black. If you look from O and the triangle you see is
white then the volume of its tetrahedron is positive, if the triangle
is black then the volume of its tetrahedron is negative.

It doesn't matter where O is. It can be inside or outside of the
contained space, the volume calcuulation will give the same result.

Adrian.
--
Adrian Rossiter
adr...@antiprism.com
http://antiprism.com/adrian