Percent Cylinder Occupation

[Michael J Spraggon]

The centroid - exactly. So this could be thought of as the organisms ability to raise nutrients above the ground and create new cells. A high cylinder occupation is not necessarily the same as a high centroid position - for example a cone has the same cylinder occupation as in inverted cone but its centroid is higher 3/4 of its height instead of 1/4 of its height.

This could be a useful measure of tree morphology.

Michael

[Michael J Spraggon]

I meant the inverted cone's centroid is higher!

[edfrank]

Matt and Michael,

The concept is sound, but we are not dealing with cones. We are talking frustums of cones - essentially the cone with the top cut off. So the center of mass is only slightly below the center height, not at 1/3 the length. The smaller the difference between the upper section diameter and that of the lower diameter, the closer the center of gravity will be to the halfway point. I can't do the math, or at least not without pain, as it is not my strength, but if nobody else steps up I will give it a shot later.

Edward Frank

[Michael J Spraggon]

I wasn't wanting to get caught up in maths at this point either. I was just thinking that if some trees would have more of their volume high up than others and that if it were calculated then the data might be useful. Photo mapping might be the easiest way to do it.

Michael

[dbhguru]

Michael and Ed,

I'll make a post in a couple of days on the topic of computing the centroid of a cone and frustum of a cone. The Internet is loaded with material on the subject, some of it very good and some of it very confusing. I hope my upcoming post won't add to the confusion, but it will be oriented to tree trunks.

Bob

[fooman]

Bob,

Don't reinvent the wheel, let Wolfram do it for you!

Conic frustum formulae:

http://mathworld.wolfram.com/ConicalFrustum.html

Paraboloid:

http://mathworld.wolfram.com/Paraboloid.html

etc...

Ed,

The other thing to use to give more resolution to the descriptive number would be the ratio of centroid height to that of a cylinder, e.g. cone to cylinder has ratio of 0.25 to 0.5, or 1/2. Non-tapering cylinder has ration of 0.5:0.5 or 1. This factor could be used to modify the standard height/girth measurements to provide a representation of the "bigness" of the tree.

Cheers,
Matt

[dbhguru]

Matt,

Good point. You used your head. Wolfram is awesome.

Strangely, I enjoy deriving the equations (masochism to many minds) to keep my aging brain from completely atrophying (alas, a losing battle), but By Jove, I did it. I'll post the derivation in the future just for the heck of it.

Here is an image that shows the centroid and associated formula for the complete conoid form and frustums thereof. I show a frustum. The conoid family includes cylinders, which can be though of as a frustum of a cone which has equal upper and lower diameters.

Bob