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Frustum Volume Equations

Posted: Sun Mar 04, 2018 7:49 pm
by dbhguru
Ents,

For trunk volume modeling, our challenge is to choose the right frustum as we work our way up the trunk. Near the base, we often use the neiloid form, followed by a paraboloid section, and ending with a cone. This is the classic forestry model. However, conical frustums usually fit short sections well enough, but our desire is to use long frustums where we can to avoid too much measuring. The formulas below cover the paraboloid, cone, neiloid, and a general formula, which is dependent on the value of the exponent p. The variable h0 = frustum height. The paraboloid, cone, and neiloid are all derivable from this general equation. The values of p are: p = 0.5 for a paraboloid, p = 1 for a cone, and p = 1.5 for a neiloid. If you wanted a form intermediate to say a paraboloid and cone, you might set p = 0.75. For a form between a cone and neiloid, you might use p = 1.25.
FrustumVolumeEquations.png
FrustumVolumeEquations.png (27.87 KiB) Viewed 908 times
In a few days, I'll submit an Excel workbook that allows you to choose the type for each frustum to include allowing you to specify a value for p if you elect to not use the paraboloid, cone, or neiloid for a frustum.

I'm doing this principally in support of the NTS A-Team of mad modelers, namely: Erik Danielsen, Elijah Whitcomb, Jared Lockwood, Dale Luthringer, Larry Tucei, and Michael Taylor (although Michael can develop this stuff in his sleep when he has the time).

Bob

Re: Frustum Volume Equations

Posted: Mon Mar 05, 2018 10:45 am
by Larry Tucei
Bob- Thanks for all you are doing to make it easier for us commoners. I will be putting it to use soon on one of larger Live Oaks in Louisiana at Oak Alley. I would estimate this tree will between 3000-4000 cubes. Larry

Re: Frustum Volume Equations

Posted: Mon Mar 05, 2018 12:18 pm
by Erik Danielsen
That general equation is a great addition! Thanks Bob.

Re: Frustum Volume Equations

Posted: Mon Mar 05, 2018 4:10 pm
by dbhguru
Erik,

I just sent you and the others an Excel workbook to calculate volume using the general volume equation. It is set up to allow you to calculate frustums for a limb and trunks/limbs for the tree, i.e. it is whole tree oriented.

The workbook is attached for all to use. I apologize in advance if it is not clear. If any of you need more explanation, let me know what is unclear and I'll work on it.

Bob