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TOPIC: Looking at the paraboloid more closely
http://groups.google.com/group/entstrees/browse_thread/thread/670710877874a342?hl=en
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== 1 of 1 ==
Date: Sun, Dec 2 2007 8:44 pm
From: dbhguru
ENTS,
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.
Let:
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
Then:
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 email. 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?
Bob
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TOPIC: TaperVolumeFrustum
http://groups.google.com/group/entstrees/browse_thread/thread/66d8d1aea55e4d08?hl=en
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== 1 of 1 ==
Date: Sat, Dec 15 2007 9:46 pm
From: dbhguru
ENTS,
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*[(Hh)/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.
where:
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}]*[(Hh2)^(2p+1)  (Hh1)^(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 h2h1 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.
Bob
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TOPIC: TaperVolumeFrustum
http://groups.google.com/group/entstrees/browse_thread/thread/66d8d1aea55e4d08?hl=en
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== 1 of 2 ==
Date: Sun, Dec 16 2007 4:59 am
From: "Gary A. Beluzo"
Bob:
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.
Gary
== 2 of 2 ==
Date: Sun, Dec 16 2007 9:45 am
From: dbhguru
Gary,
Sure. Let's go for it.
Bob
