==============================================================================
TOPIC: Volumes 101
http://groups.google.com/group/entstrees/browse_thread/thread/f142ddf5cfa2b5e3?hl=en
==============================================================================
== 1 of 1 ==
Date: Thurs, Nov 22 2007 6:07 am
Bob Leverett
ENTS,
The recent deluge of volume equations by yours truly is but a
prelude to some serious big tree modeling of white pines,
hemlocks, sycamores, sugar maples, tuliptrees, etc. by Will Blozan,
Gary Beluzo, and me. Hopefully others of you will join us to make
this a serious effort. In particular, I hope Dale Luthringer,
Howard Stoner, and John Eichholz can spare us some time. Ed Frank
will certainly have a place, which I leave for our distinguished
webmaster to define. I don't speak for Lee Frelich Don Bragg, Tom
Diggins, or BVP. But, I would think that one or more will fit in
after the ground troops have accumulated a sizable amount of data
and individual modelings for them to consider for potential
scientific papers. They are the participating ENTS heavyweights.
The end result will be more than just volumes for the selected
trees. We should gain numerical insights into the way each species
accumulates wood as it progresses from young to mature to old
growth.
The dendromorphometry project that I've assigned to myself has
occupied a lot of my time as of late. That's okay since I can't
get outside except for brief periods and must remain relatively
close by. So the idea hit me to summarize what we have in the way
of forms and formulas to model trees in a series of emails that
will likley put the most stubborn insomniac into a deep and
restful sleep. But seriously, I hope those of you who are
interested in the numeric side of tree forms will take these
submissions seriously. I'll begin with the right circular cone. www.mathwords.com
<http://www.mathwords.com/> gives key formulas
relating to the cone, but we're going to extend what that source
offers.
Right Circular Cone.
The right circular cone is the form we commonly employ when
modeling tree trunks. The base of the cone is a circle. Its sides
are converging straight lines coming to a point directly above the
center of a circle. A perpendicular line from the apex to the base
intersects the center of the circle. Let's reduce this idea to
mathematics.
Let:
H = height of cone
R = radius of the circular base
V = volume of cone
pi = 3.141593
Then:
V = (1/3)*pi*R^2*H
If the tree is a circular cone, but not right circilar, the
volume formula is the same. Circular cone trees that aren't right
circular cones are leaners.
In actuality, few trees have trunks that are shaped like a cone
extended from base to tip of the trunk. However, sections of the
trunk may be conical, and the shorter the section of trunk, the
more it can be modeled by a section of the cone created by passing
two parallel, horizontal planes through the cone. Being
horizontal, the planes are at right angles to the center line of
the cone. The result of the bisecting planes is called a frustum
(not frustrum). The value of the frustum to us has been enormous.
It can't be overstated. So, let's now look at two formulas that
give the volume of a frustum of a cone. We begin with definitions.
Let:
pi = 3.141593
r1 = radius of the circle forming the base of the cone.
r2 = radius of the circle forming the top of the cone
d1 = diameter of the circle forming the base of the cone.
d2 = diameter of the circle forming the top of the cone
c1 = circumference of the circle forming the base of the cone.
c2 = circumference of the circle forming the top of the cone
h = height of cone; vertical distance between base and top of
frustum
h1 = height of the circle forming the base of the frustum above
the base of the cone
h2 = height of the circle forming the top of the frustum above
the base of the cone
R = radius of base of cone
D = diameter of base of cone
C = ciircumference of base of cone
H = height of cone.
f1 = proportion of volume based on height; f1 = 2/3rd
represents a point at 2/3H as measured from the apex.
V = volume of frustum of cone
Then:
V = [(pi*h)/3]*[r1^2 +r1*r2+r2^2] volume of frustum based
on radius
or
V = [(pi*h)/12]*[d1^2 +d1*d2+d2^2]
or
V = [(pi*h)/12]*[c1^2 +c1*c2+c2^2]
These3 formulas are equally valid. One can easily be derived
from another. The choice of which one is most convenient to use is
situtionally dependent. The above formulas assume a cone, but do
not require that the cone be part of a super cone that encircles
the entire trunk. Each frustum can represent a different cone. The
section of the tree being modeled happens to conform to the parent
cone for at least the length of the frustum.
Suppose we want to know how much volume lies between heights h1
and h2 that delineate the frustum. We can't get the answer from
the preceding formulas. We need a new formula and that formula is:
V = (pi/3)*(R/H)^2*[h2^3h1^3],
where h1 and h2 are measured from the apex of the cone toward
the base. If the measurement is up from the base, the formula
becomes:
V = (pi/3)*(R/H)^2*[(Hh2)^3(Hh1)^3].
If we want to express some height h in terms of % of cone
height H, measured from the apex toward the base, then we can
compute the volume contained in the portion from the apex to
height h. If f1 is a proportion such as 1/4th, 1/2, 2/3rds, etc.,
then the volume occupied between the apex and h (f1*H) is
given by:
V = (pi/3)*(R^2)*H*f1^3.
This last formula has many potential uses. For example, suppose
we want to know the volume at 50% of the height H of the cone from
the apex downward. Then f1=1/2 and f1^3=1/8. We
susititute 1/8 in the above equation along with the basal radius
and height to get V.
Note that in the above formula, (pi/3)*(R^2)*H is
just the volume of the cone. To avoid confustion, assume we define
the full cone volume as Vc = (pi/3)*(R^2)*H. Then in the
above formula for V, we can say V = Vc*f1^3. If we divide
both sides by Vc, we get V/Vc = f1^3, which gives the
proportion of the cone's volume (as opposed to the actual amount)
that lies with in f1 of H. We only have to calculate f1^3 to get
the proportion of the total volume that falls within f1 proportion
of the cone based on height. This is a surprisingly simple
calculation. What insights does this afford us? If we had a
cylinder, then the first 1/3rd of the cylinder based on its height
would contain 1/3rd of its volume. The first 1/2 of the cylinder
would contain 1/2 the volume abd so on. Bit in the case of the
cone, its is very different. For f1=3, f1^3 = (1/3)^3 =
1/27. It may come as quite a surprise that the first 1/3rd of
a cone based on height conta ins only 1/27th the volume. The
following extract table from an Excel spreadsheet shows just how
unexpected the relationship between % of total height and % of
total volume for a cone is.
Profile of Accumulating Cone Volume against proportion
of total height
















f1

H

h

R

r

Va

V

Va/V

0.1

100

10

3

0.3

0.94

942.48

0.10%

0.2

100

20

3

0.6

7.54

942.48

0.80%

0.250

100

25

3

0.8

14.73

942.48

1.56%

0.3

100

30

3

0.9

25.45

942.48

2.70%

0.333

100

33.3

3

1.0

34.80

942.48

3.69%

0.4

100

40

3

1.2

60.32

942.48

6.40%

0.5

100

50

3

1.5

117.81

942.48

12.50%

0.6

100

60

3

1.8

203.58

942.48

21.60%

0.667

100

66.7

3

2.0

279.67

942.48

29.67%

0.7

100

70

3

2.1

323.27

942.48

34.30%

0.750

100

75

3

2.3

397.61

942.48

42.19%

0.8

100

80

3

2.4

482.55

942.48

51.20%

0.9

100

90

3

2.7

687.07

942.48

72.90%

1.0

100

100

3

3.0

942.48

942.48

100.00%









Definitions:








f1 = proportion of total cone height H








H = total cone height








h = height of proportion f1








R = radius of cone








r = radius of proportion








Va = volume of proportion








V = volume of cone








Va/V = proportion of cone's total volume for f1








To reach 50% of the volume of a cone, the height (depth) from
the apex can be determined as follows:
0.50 = f1^3
0.50^(1/3) = (f1^3)^1/3
f1 = 0.7937
This means that from the apex you must travel 79.37% of the
height of the cone to reach 50% of the volume. This suggests that
the form that most tree trunks take isn't that of one encompassing
cone. No surprises there, but frustums of cones are extremely
valuable to us. Can we relate a particular frustum to the parent
cone? How does one go about computing the full height of the cone
that has as a frustum the one being modeled on a tree trunk?
Calculating the rate of taper is the answer.
Let:
d1 = diameter of top of frustum
d2 = diameter of base of frustum
ho = height of frustum
h = height of cone above frustum
h1 = length of tree below frustum to base
D = measured diameter of base
Dc = computed diameter of base by extension of frustum
H1 = actual height of tree above frustum
Then:
h = d1/[(d2d1)/h0]
The height of the column including the frustum is just: h =
d1/[(d2d1)/h0] + h0. The full cone constructed from the
frustum would be:
h = d1/[(d2d1)/h0] + h0 + h1.
Does the extension of the frustum to the base of the tree match
the measured diameter D?
Dc = h1*[(d2d1)/h0] + d2
The degree to which the frustum represents a cone that matches
the trunk can be tested by two measurements: (hH1) and (DcD).
If the constructed cone matches the trunk, then these differences
will be small.
Readers should remember that the trunk of a tree seldom can be
accurately modeled with a single geometric solid. Use of the
frustum to model the trunk by sectioning the trunk at places where
curvature appears to change is the best way. Trying to construct
exotic polynomials or exponential or logarithmic equations are
exercises in academic silliness.
The next email will explore use of paraboloid frustums and
testing for the paraboloid shape. We will eventually see that
there is no single, encompassing equation that will represent most
tree trunks. We will also see that the beyond the root flare, the
form of the trunk will usually fall between a paraboloid and cone.
Moee exotic curve forms are mostly academic playing around and not
worth the effort expended.
Bob
Volume 101  3
ENTS,
In lesson #1 of Volumes 101, I presented the
right circular cone as one of the most important geometric solids
for modeling tree trunks. I emphasize that the entire trunk can
seldom be modeled by a single geometric solid because the trunk
changes curvature at different points. Some parts of the trunk may
be conical, but other parts may take on the shape of a an
important geometric solid called a paraboloid. A paraboloid
inscribed in a cylinder will occup half the space of the cylinder
as opposed to a cone, which occupies a third of the space of the
cylinder. A paraboloid can be generated by rotating one half of a
parabola around an axis through the apex. A parabola is a second
degree curve of the form y = a*x^2 +b*x*y + c*y^2+d. However, for
our purposes there is a much simpler form of the general parabola
equation. First, some definitions.
Let:
R = radius of paraboloid
H = height of paraboloid
r = radius at an arbitrary point on the
paraboloid
h = height of frustum
h1 = height to base of a frustum of the
paraboloid starting at the apex
h2 = height to top of a frustum of the
paraboloid starting at the apex
r1 = radius of paraboloid at point h1
r2 = radius of paraboloid at point h2
Vp= volume of frustum
V = volume of entire paraboloid
pi = 3.141593
Then:
r = R*SQRT(h/H) is the equation of a
parabola. Or stated in terms of h, we get: h
= (H/R^2)*r^2.
Since we're seldom interested in a paraboloid
that included the entire trunk, we need a formula for the frustum
of a paraboloid. Two formulas are presented below.
Vp = (1/2)*pi*(h2h1)*(r1^2 + r1r2 +
r2^2)
or
Vp = (1/2)*pi*R^2/H*(h2^2  h1^2)
Note that if the height of the frustum is
measured independently and without being referenced to the apex,
then the first formula can be changed to:
Vp = (1/2)*pi*(h)*(r1^2 + r1r2 + r2^2) and
it has as its conoid counterpart the formula Vp
= (1/3)*pi*(h)*(r1^2 + r1r2 + r2^2).
In future trunk modelings, these two formulas
will likely be duking it out. So to choose between them, we will
need to devise tests, which will be the subject of the 3rd
installment of Volumes 101.
Bob
ENTS,
The forms taken by tree trunks seldom fit either a
cone or a paraboloid. Most of the time they fall in between the
two forms. One way to see how well a form fits is to postulate an
overall form and then test to see what the model gives at points
where actual girth measurements have been taken.
Let:
R = radius of base of a paraboloid and cone matching
the trunk at 4.5 feet.
H = full height of tree for modeling purposes
h = height of tree at a point being checked.
Height can be determined from either the base or apex of the
solids.
r = computed radius on model at height h
f = weight given to paraboloid form where
0<=f<=1
Then:
If measuring from base
r = (R/H)*[f1*SQRT((Hh/H)) + (1f)*((Hh)/H)]
If measuring from the apex
r = (R/H)*[f1*SQRT((h/H)) + (1f)*((h)/H)]
These formulas will be frequently applied in the modelings to
come. At this point for the OG hemlocks the best fit is for f
= 0.67.
ENTS,
To make some of these discussions a little more
concrete, suppose we have two trees. Each tree's trunk is 4 feet
in diameter and 100 feet tall. One trunk is a perfect cone and one
trunk is a perfect paraboloid. The following table profiles the
radii of each trunk going from apex to base. It shows how the
radius changes under the assumption of paraboloid and cone.
Bob

R= 
2 
Hgt= 
100 
%
Hgt 
Hgtft 
Para
r 
Coner 

10.0% 
10 
0.63 
0.20 

20.0% 
20 
0.89 
0.40 

25.0% 
25 
1.00 
0.50 

30.0% 
30 
1.10 
0.60 

33.3% 
33.3 
1.15 
0.67 

40.0% 
40 
1.26 
0.80 

50.0% 
50 
1.41 
1.00 

60.0% 
60 
1.55 
1.20 

66.7% 
66.7 
1.63 
1.33 

70.0% 
70 
1.67 
1.40 

80.0% 
80 
1.79 
1.60 

90.0% 
90 
1.90 
1.80 

100.0% 
100 
2.00 
2.00 

Dale, Will, Lee, Don Bragg, Gary, Jess, Ed, et al:
The attached Excel workbook represents the
latest incarnation in my attempt to provide a useful tool for
trunk modeling that builds on our already extensive
experience by adding some tools to analyze form. The
workbook consists of 3 spreadsheets, a main spreadsheet, an
instructions spreadsheet, and an example spreadsheet. The main
spreadsheet allows the user to volume model the trunk of a tree.
More specifically, the main spreadsheet allows the user
to enter: (1) trunk measurements from base to some
point up the trunk (usually the top), (2) the tree's
total height and diameter at 4.5 feet, and (3) an optional
height and diameter at a point up the trunk that the user
considers important for analyzing the overall shape of
the trunk from that point down. For example, maybe the first 100
feet of a 130foot tree appears to be in the shape of a paraboloid and
the remaining 30 more in the order of a cone. We might want
to investigate this possibility. It could influence how
we treat the individual frustums that represent the
measurements.
In the main spreadsheet, the light blue cells
are for user input. Everything else is represents titles or
is calculated. The cells containing titles have
a salmoncolored background color. The calculated cells are
sandcolored. The spreadsheet has the protection feature invoked
(without password) to prevent inadvertant erasing or overwriting a
cell with formula in it. That is a common occurrence
especially when dealing with a spreadsheet someone else created.
In row #1, the user enters the tree name,
species, and location. On the 2nd row, the user enters the
tree's full height, DBH, and optionally, the height of a
"trial" spot on the trunk, below which the user believes
can be represented by a single geometric solid. The diameter is
then entered for the trial height. Finally, a tolerance number is
entered, which will allow a deviation between the actual measured
diameters and theoretical ones based on the conoid model.
Exceeding the deviation suggests the likelihood of either a
paraboloid or neiloid. The value 0.03 is a good starter, i.e. a 3%
deviation.
After the 2 header rows, data entry proceeds
column wise. Individual points of trunk measurement are
entered in column A as total height above the base. The
heights of the individual frustums are automatically calculated in
column B from the running or cumulative height of column
A. Column C holds the measured diameters at the height points
of column A. Column D is the radius automatically calculated from
the user entered diameters of column C.
The measurements entered in columns A and C
generate a series of adjacent, or stacked, frustums of
geometric solids. To calculate frustum volume, the user must
indicate the type of frustum by entering in column E either P for
paraboloid, C for cone, or N for neiloid. The height,
diameter, and type frustum information accounts for columns AE.
Columns FH automatically complete the volume calculations.
Columns IQ help to identify for the user the best choice of
frustum type by providing interpolated diameters based on:
(1) the immediate area around a measurement, (2) the entire
trunk, and (3) a significant segment of the trunk, e.g. the
first 75 feet.
To analyze the trunk shape in the vicinity of a
particular diameter measurement, the preceding and following
diameter measurements are used to create a frustum that includes
the target measurement as an intermediate point. The
theoretical diameter at the particular measurement is calculated
by interpolation based on a conical form. The interpolated
diameter is then compared to the actual one which was
entered in column C. If the actual measurement is
within a target percent, e.g. 3%, of the
interpolated measurement, then the suggested shape at the target
measurement is a cone. If the actual measurement is less
than the interpolated value by more than the target percent, the
form is assumed to be neiloid. If the actual measurement exceeds
the interpolated measurement by more than the target percent, the
paraboloid is suggested as the shape at the point under
consideration. Note that this is a localized fo rm. Each
frustum measurement is analyzed similarly. But what about the big
picture? Does the entire trunk look conical or paraboloid? Taking
the DBH and full height of the trunk, a cone and a
paraboloid are considered in columns LO. In this big picture, the
diameter at each point of measurement is compared to both the
corresponding diameter of an encompassing cone and of
the paraboloid. The differences are noted. Columns LO contain
these measurements with comparisons to the actual. All these
columns are automatically calculated
If the user supplied a trial height and
diameter on row #2, then a long frustum is assumed from the
trial height down to the base. The measured
diameters are compared to the theoretical diameters calculated for
the long frustum, within the range of the frustum of course. This
allows the user to take a longer section of trunk and analyze
what is happening a each measurement point. The entire trunk
may not satisfy the requrements of say a single paraboloid,
but an extended section may.
In summary, the user can analyze the
information from these three levels of comparison before
settling on a C, N, or P for column E entries. Once the frustum
type is chosen, the volume of the frustum is calculated for that
type. The automatically calculated volume appears in column
G. Column H automatically accumulates the volumes of the frustums
in a running total. The user is free to experiment with the
type of each frustum based on the data from the subsequent columns
and/or experience.
No sooner than I've created a feature, I think
of improvements. However, what is being presented in the
spreadseet increases the flexibility of the modeling process.
Nonetheless, improveed version are sure to follow.
==============================================================================
TOPIC: Volumes 101  6
http://groups.google.com/group/entstrees/browse_thread/thread/1d4b3b11a90df86e?hl=en
==============================================================================
== 1 of 1 ==
Date: Wed, Nov 28 2007 8:14 pm
From: dbhguru
Will, Lee, Dale, Don, Jess, Howard, Ed, et al.:
I've been concentrating on formulas for the
frustum of a paraboloid. Here is what I've come up with. First, the
usual definitions.
Let:
H = height of paraboloid containing the frustum
R = radius of base of paraboloid containing the
frustum
h1 = height of base of frustum measured from the
paraboloid's apex
h2 = height of top of frustum measured from the
paraboloid's apex
r1 = radius of base of frustum
r2 = radius of top of frustum
pi = 3.141593
Vp = volume of frustum
Then:
1. Vp = [pi*R^2/(2*H)]*[h2^2  h1^2]
2. Vp = [pi*H/(2*R^2)]*[r2^4  r1^4]
3. Vp = (pi/2)*(r2^2*h2  r1^2*h1)
If h1 and h2 are measured from the paraboloid's base
Then:
4. Vp = [pi*R^2/H]*[(h2h1)*[H 
(h2 + h1)/2]
The formula that I've been using is:
Vp =
[pi*(h2h1)/2]*[r1^2 + r2^2 + r1*r2]
However, I've been unable to derive it. It makes sense, but I
can't seem to reduce any of the other forms to the above form. I'll
keep trying.
Bob
==============================================================================
TOPIC: Putting it all together
http://groups.google.com/group/entstrees/browse_thread/thread/4cc03aa070212404?hl=en
==============================================================================
== 1 of 1 ==
Date: Thurs, Nov 15 2007 12:55 am
From: dbhguru
ENTS,
Through out the remainder of November, I plan to bring the ENTS
dendromorphometry PowerPoint presentation up to snuff. When
complete, I'll ship it to Ed and trust to his organization and
presentation abilities to create a website version where it will be
available to everyone. I'll also ship the presnentation to Don Bragg
for his consideration.
Dendromorphometry is the art and science of measuring trees in the
field. It is the special creation of ENTS and embodies the extreme
discipline that distinguishes us in our search for truth in the
dimensions. Dendromorphometry includes the mathematical models and
formulas, the measurement protocals, and the discovered
relationships that distinguish individual species. Where there are
multiple approaches to doing a particular kind of measurment (e.g.
tree height), dendromorphometry orders them form most to least
accurate or risky and assesses the probability of significant error.
In fact, a lot of what dendromorphometry is about is identifying and
quantifying of the sources of error that can be associated with a
particular measurment technique. Dendromorphometry includes
"cookbook recipes" for carrying out measurements. Finally,
dendromorphometry investigates the distinguishing shape
characteristics of different species and reduces them to
mathematical constants and relations
hips.
It is the practice of dendromorphometry and a fanatical insistence
on achieving a high level of accuracy that distinguishes ENTS. While
we support the state and American Forests champion tree programs, we
separate ourselves from the broad, inclusive purposes of those
programs. Although we seek to recruit members who will use
dendromorphometry and add to our databases of species, we do not
reduce our expectations in order to gain recruitment.
ENTS does not promote any particular brand of equipment. It does
evaluate different brands, identifying the strengths and weaknesses
of each.
Bob
