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 e-mails 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*[h2^3-h1^3],

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*[(H-h2)^3-(H-h1)^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/[(d2-d1)/h0]

The height of the column including the frustum is just: h = d1/[(d2-d1)/h0] + h0. The full cone constructed from the frustum would be:

h = d1/[(d2-d1)/h0] + h0 + h1.

Does the extension of the frustum to the base of the tree match the measured diameter D?

Dc = h1*[(d2-d1)/h0] + d2

The degree to which the frustum represents a cone that matches the trunk can be tested by two measurements: (h-H1) and (Dc-D). 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 e-mail 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*(h2-h1)*(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((H-h/H)) + (1-f)*((H-h)/H)]

 

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	Hgt-ft	Para -r	Cone-r
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 130-foot 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 salmon-colored background color. The calculated cells are sand-colored. 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 A-E. Columns F-H automatically complete the volume calculations. Columns I-Q 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 L-O. 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 L-O 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.  

 

Bob