Looking at the paraboloid more closely Bob Leverett

TOPIC: Looking at the paraboloid more closely

== 1 of 1 ==
Date: Sun, Dec 2 2007 8:44 pm
From: dbhguru


Most of Will's, Jess's, and my trunk modeling has broken the trunk up into enough frustums to justify the use of the cone as the underlying geometric form. However, when longer trunk segments are taken, we shoul take curvature seriously. We need to be aware of important properties of at least three geometric solids used to model tree trunks - conoid, paraboloid, and neiloid. In this email, I deal with the paraboloid.

Imagine a tree trunk that looks at least passably like a paraboloid. The huge Smoky Mountain hemlocks would seem to fit the bill. If we adopt the paraboloid as a modeling form, do we attempt to place the entire trunk under the control of a single base to tip paraboloid? Or do we see a paraboloid starting at BH and moving up to some point on the trunk below the top? Or do we see segments of the trunk under the control of different paraboloids? Well, it can be any of the three. So we will eventually need to investigate each modeling scheme.

The basic formula for the paraboloid is (1/2)*area of base*height. However, it is almost guaranteed that we would never find a tree trunk that would hold the shape of a single paraboloid from base (or BH) to tip. We would expect to deal with frustums of one or more paraboloids. But, let's begin with a single paraboloid and deal with frustums of that one geometric form even if we never intend to use it in its entirety. For instance, we might expect to apply the paraboloid form to the lower 2/3rds of the trunk. But to get to that region of the trunk, we might still have to measure down from the apex.

Starting from the apex and measuring toward the base, the volume of a frustum contained between points representing heights h1 to h2 from the apex can be computed as explained below.


H = total paraboloid height
R = radius of paraboloid at base
h1 = point at base of frustum as measured from apex
h2 = point at top of frustum
h = arbitray point at distance h from apex
Vp = volume of frustum
V = volume of paraboloid
pi = 3.141593
p = proportion of V that occurs between the apex and h


V = (pi*R^2*H)/2 (Entiee volume)

Vp = [(pi*R^2)/(2*H)]*[h2^2 - h1^2] (Volume of frustum)

Supppose we would like to know at what h value a total of p% of V occurs? Since we start at the apex, we can use the formula for Vp to represent the volume between the apex and h.

h1 = 0, since we're starting at the apex.

Vp = [(pi*R^2)/(2*H)]*[h^2] where h = h2.

By definition, p = Vp/V. Substituting, we get

p = [(pi*R^2)/(2*H)]*[h^2] / (pi*R^2*H)/2 = h^2/H^2

Therefore, h = H*SQRT(p)

From the last equation, we can form the ratio h/H = SQRT(p). Now suppose h/H = 0.5, i.e.h is 50% of the total height of the paraboloid. Then 0.5 = SQRT(p) and p = 0.25. This says that for a paraboloid form based on the equation r = R*SQRT(h/H), 25% of the volume occurs in the first 50% of the height of the paraboloid when starting at the apex and moving toward the base. This means that the bottom 50% of the paraboloid holds 75% of the volume - if the paraboloid fully fits the trunk.

If we start measuring height upward from the base, the equations become surprisingly messy. So I chose to go from top down. However, it is likely that we would be going from base toward apex. I'll deal with this situation in another e-mail. My point in presenting this analysis is to focus attention on how these geometric solids accumulate volume. How far do we have to go up a narrowing trunk before we can saefly conclude that whatever is left is minimal?


TOPIC: Taper-Volume-Frustum

== 1 of 1 ==
Date: Sat, Dec 15 2007 9:46 pm
From: dbhguru


As part of the paper that Gary Beluzo, Will Blozan, and I are preparing for the next issue of the Bulletin, I set out to derive a general taper equation that would give rise to the volume formulas of the three common geometric solids that we use (cone, paraboloid, neiloid). However, being general, the taper equation would yield volume formulas intermediate to the big three, plus extend the field to forms more concave than the neiloid or convex that the paraboloid. Well, the deed is done. The general taper equation is.

r = R*[(H-h)/H]^p

where r = radius of the solid at height h above the base of radius R. Total height of the solid is H and p is the all important taper parameter. From the taper equation, we can derive an associated volume equation for each value of p. The generalized volume equation is:

V = [1/(2*p+1)]*pi*R^2*H.


V = volume
pi = 3.141593
R = radius of base of solid
H = height of solid
p = taper parameter

From the above two equations, we can derive a third equation for a frustum of the solid. The equation is:

V = -[(pi*R^2)/{(2p+1)*H^2p}]*[(H-h2)^(2p+1) - (H-h1)^(2*p+1)].

where h1 and h2 represent heights above the base that define the frustum. The variable h1 represents the height of the base of the frustum above the base of the solid and h2 is the height of the top of the frustum above the base of the solid. The quantity h2-h1 is the height of the frustum.

If p = 1, we get the cone. If p = 1/2, we get the paraboloid. If p = 3/2, we get the frustum. A value of p= 3/4 represents a taper halfway between the cone and the paraboloid. This family of three equations is going to get a thorough workout in future trunk and limb modeling exercises.

There is nothing that instructs tree trunks to form themselves into convenient solids such as cones, paraboloids, and neiloids for a personal computational convenience. The large reservoir of tree trunk data that ENTS is building, courtesy of Will's climbing feats has produced the data necessary to allow us to really analyze trunk taper. With the 3 generalized equations given above, we are no longer prisoners of the big three.


TOPIC: Taper-Volume-Frustum

== 1 of 2 ==
Date: Sun, Dec 16 2007 4:59 am
From: "Gary A. Beluzo"


In addition to the Index of Dendricity we could also use the trapezoid
(truncated cone) shape for modeling. It is used to approximate the
morphology of the lake basin in limnological modeling. Let me know if
you want me to post it. Hell, we should just check out all of the
solid geometry equations because sooner or later we will find a tree
that fits the curve.


== 2 of 2 ==
Date: Sun, Dec 16 2007 9:45 am
From: dbhguru


Sure. Let's go for it.