### Frustum Volume Equations

Posted:

**Sun Mar 04, 2018 7:49 pm**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.

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

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.

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