| Useful
Formula List |
Robert
Leverett |
| Apr
19, 2007 11:10 PDT |
ENTS,
As a consequence of the
mathematics that I've been wading through
as of late, it occurred to me that a useful byproduct of the
effort
might be a listing of our most frequrntly used formulas and
measurement
processes, a top ten list or something along those lines. Well,
as a
first crack, the simple H = Lsin(A) leads the pack, where A is
the angle
from the eye to the target and L is the straightline distance
from eye
to target. Of course, we use this formula twice to compute the
above and
below eye level components of a tree's height.
In terms of frequency of usage, D
= Lcos(A) might be second, where
D is the straightline, eye level distance to the point
vertically below
(or above) the target point. This frequently represents the
distance to
the trunk of a tree at eye level.
I have to throw in the tangent
formula out of recognize for its
value in places where it fits: H = Dtan(A), where H and D are
defined
as above.
A 4th formula of great use to me
is A^2 = B^2 + C^2 - 2*B*C*cos(a)
or the law of cosines. It allows us to compute the straightline
distance
between two points in space using a triangle connecting the two
points
and our eye. B and C are the distances to the two points from
our eye, a
is the angle between B and C, and A is the connecting distance
between
the two points.
A 5th formula stated somewhat
generally is any algebraic derivation
of a/b = A/B. This formula is expresses the proportionality of
sides for
similar triangles.
Number 6 relates circumference (C)
to the diameter (D) of a circle:
C = PI*D.
Number 7 is the formula for the
cross-sectional area of a circle: A
= PI*R^2.
Number 8, the elliptical
equivalent is A = PI*a*b, where a and b
are the semi-major axes of the ellipse. Going from two to three
dimensions, the most useful volume formula for us is the one for
the
frustum of a cone.
So, #9 is: V = H/3*(A1 + A2 +
SQRT(A1*A2) where H is the height of
the frustum, and A1 and A2 are the cross-sectional areas of the
top and
bottom of the frustum.
Finally, I would include as
#10,the Pathagorean relation for the
length of the hypotenuse of a right triangle in terms of the
lengths of
the other two sides. If C is the hypotenuse and A and B are the
other
two sides, then we have C^2 = A^2 + B^2.
So, let's now summarize these 10
formulas, noting the definitions
of the individual variables for each formula as defined above.
1. H = Lsin(A)
2. D = Lcos(A)
3. H = Dtan(A)
4. A^2 = B^2 + C^2 - 2*B*C*cos(a)
or A = SQRT[A^2 + + C^2 -
2*B*C*cos(a)]
5. a/b = A/B
6. C = PI*D
7. A = PI*R^2
8. A = PI*a*b
9. V = H/3*(A1 + A2 + SQRT(A1*A2)
10. C^2 = A^2 + B^2
I am taking onto myself the
project of keeping a running list of
useful formulas to ENTS work for periodic dissemination through
our
e-mail list and on the website. I'll work out a better format
for future
submissions, one that is consistent and always defines the
variables
being used in the formulas.
Bob
Robert T. Leverett
Cofounder, Eastern Native Tree Society
|
| RE:
Useful Formula List |
Paul
Jost |
| Apr
20, 2007 18:33 PDT |
Bob,
4 and 10 are really the same equation with 10 being a special
case of 4 with
a term eliminated at 90 degrees.... The Pythagorean theorem is
really just
a special case of the law of cosines applied to a right
triangle...
PJ
|
| RE:
Useful Formula List |
Robert
Leverett |
|
Apr
23, 2007 06:50 PDT |
Paul,
Yes, I understand. For others on the list who
may not see this,
C^2 = A^2 + B^2 - 2*B*C*cos(c) Law of
Cosines
C^2 = A^2 + B^2 Pathagorean
relation
for
right triangles
If c = 90 degrees, then cos(c) = 0
and through substitution of the latter into
C^2 = A^2 + B^2 - 2*B*C*cos(c)
we get, C^2 = A^2 + B^2 - 2*B*C*cos(90)
This then simplifies to:
C^2 = A^2 + B^2 - 2*B*C*0
and finally to
C^2 = A^2 + B^2
Paul, like almost all electrical
engineers is extremely strong in
mathematics. He was the first of us to make use of the law of
cosines to
check on tree height calculations. Thanks, Paul. I have a
feeling that
if I make a mistake in my derivations, you'll be among the first
to catch
it.
Bob
|
| 10
going to 11 and beyond |
Robert
Leverett |
| Apr
23, 2007 10:13 PDT |
ENTS,
Exploring the features of
trees in the field mathematically and discovering dimensional
attributes
that may be obvious only in hindsight is where we are headed.
The
challenge is to bring our whole group along. The ENTS motto
should be:
Leave no Ent Measurer Behind".
However, the discovery
mission just articulated does not mean that
we will need to launch a "new formula of the week"
program. Our progress
needs to be methodical and practical and that was the purpose of
the big
ten list. What should be #11? The formula that I propose is the
one to
measure tree height using the tangent function and two vantage
points as
exhibiting significant potential, but not yet proven
practically. The
formula referred to is:
H = [Dtan(A)tan(B)]/[tan(A)-tan(B)].
This formula assumes that we
have lined up the point being
measured in the crown with two vantage points on the ground,
i.e. the
three points fall within the same vertical plane. We take the
horizontal
distance at eye level between the two vantage points as D and we
measure
the angles of the crown point and eye level from the two vantage
points
as A and B and then apply the above formula. Jess Riddle
believes that D
must be at least 20 feet to get sufficient differentiation for
angle
that are otherwise close together in value. I would bet that he
is
right. So, I will add the above formula to the previous list of
10, duly
noting Paul's point that 10 is really a variation of 4. I've
employed
the computer symbols for math operations to make the formulas a
little
clearer.
BIG ELEVEN:
1. H = L*sin(A)
2. D = L*cos(A)
3. H = D*tan(A)
4. A^2 = B^2 + C^2 - 2*B*C*cos(a)
or A = SQRT[A^2 + + C^2 - 2*B*C*cos(a)]
5. a/b = A/B
6. C = PI*D
7. A = PI*R^2
8. A = PI*a*b
9. V = H/3*(A1 + A2 + SQRT(A1*A2)
10. C^2 = A^2 + B^2
11. H = [D*tan(A)*tan(B)]/[tan(A)-tan(B)].
Unless someone else sees it differently, the
two variations of #11
for handling situations where the baseline is not level, but
slanted,
will not be added to the list until #11 proves its value. City
parks
with big spreading trees that can be seen from a distance on
grassy
fields should provide a satisfactory testing ground for formula
#11. If
they don't we are positioned to lay down and take a nap.
Bob
Robert T. Leverett
Cofounder, Eastern Native Tree Society
|
| 12-14
for your viewing pleasure |
Robert
Leverett |
| Apr
25, 2007 06:56 PDT |
ENTS,
The parade of formulas continues. Since
we, in ENTS, use the sine
top-sine bottom formula primarily to measure tree height, a
fairly
simple formula that reflects the sensitivity of changes in
height to
changes in hypotenuse distance, angle, or both would be
extremely
useful. One can attack the problem directly and compute the
change in
tree height based on a direct application of the sine formula to
the new
values and subtract the sine application to the old values to
get the
change or differential. The following development shows that
result.
Let:
L =
original measured hypotenuse distance to target
j = a
small change in distance
A =
original measured angle to target
k = a
small change in angle
H1 =
height based on L and A
H2 =
height based on L + j and A + k
Then:
H1 = L*sin(A)
H2 = (L +
j) * sin(A + k)
h =
H2 - H1
Or stated as a single
formula:
h = (L +
j) * sin(A + k) - L*sin(A)
This formulaic statement of the height
difference is fairly obvious.
However, another way of approaching the problem is through total
differentials, a concept in calculus that computes the change in
a
dependent variable associated with small changes in one or more
independent variables:
dH = sin(A)*dL +
L*cos(A)*dA,
where
dH = change in H
associated with,
dL,
the change in L, and
dA,
the change in A.
From the meaning of H2-H1, i.e. the full
change in height, we
defined h. We acknowledge dH to be an approximation of h. But
the
difference between h and dH is miniscule for relatively small
values of
dL and dA. So, the differential dH can be used to estimate the
error in
H from being a little off in distance or angle or both. Please
keep in
mind that in the dH formula, angles are measured in radians - an
angular
measure that represents the ratio of the length of an arc of a
circle to
the radius of that circle. There are 2 * PI radians in a circle.
Therefore, one radian = 360/(2*PI) = 57.29578 degrees. To
convert
degrees to radians, use the simple formula:
R
= A * (PI/180)
where A = the angle
expressed in degrees
PI
= 3.1415926
R
= the angle expressed in radians.
I especially like the differential
formula dH because it highlights
the quantities dL and dA and their respective roles. You can
easily
trace the separate effects of the angle and hypotenuse changes
for
different values of L and A.
In terms of usefulness, or at least
potential usefulness, I would
add the above formulas to our growing list, which presently
stands at
11. To summarize the new formulas:
12. R =
A *(PI/180)
13. A =
R*(180/PI)
14. dH
= sin(A)*dL + L*cos(A)*dA
I note that the derivation of dH is a
straightforward application of
the definition of a total derivative applied to the formula: H =
L*sin(A). I would include the simple derivation here, but
notation of
partial derivatives or differentials is a problem in Topica’s
minimal
editor.
I will pass with this explanation of
what the recent deluge of
formulas means. The intensity of the mathematics isn’t a new
devotion
for me. Mathematical modeling, practiced at one level or
another, goes
on all the time in my noodle. It was my bread and butter while
still in
the U.S.A.F and at the Pentagon. With no false modesty, my
penchant for
math and math modeling is where the ENTS approach to sine-based
mathematics originated. But until recently, I’ve hesitated to
put much
formula derivation into these e-mails with the exception of
sine-based
measuring along with a scattering of other stuff. I’ve been
afraid of
turning off the Ents who are not into the mathematics of tree
measuring.
However, I realize that those of you who aren’t oriented in
that
direction will just skip the math intensive e-mails. However, to
give
initial warning, I will supply e-mail titles that communicate
their
contents as that of tree measuring.
So here is a recapitulation of the full
list of hot-to-trot
formulas.
1. H = L*sin(A)
2. D = L*cos(A)
3. H = D*tan(A)
4. A^2 = B^2 + C^2 - 2*B*C*cos(a)
or A = SQRT[A^2 + +
C^2 - 2*B*C*cos(a)]
5. a/b = A/B
6. C = PI*D
7. A = PI*R^2
8. A = PI*a*b
9. V = H/3*[A1 + A2 + SQRT(A1*A2)]
10. C^2 = A^2 + B^2
11. H = [D*tan(A)*tan(B)]/[tan(A)-tan(B)].
12. R = A *(PI/180)
13. A = R*(180/PI)
14. dH = sin(A)*dL + L*cos(A)*dA
My current guess is that the full list
of formulas most useful to
our tree measuring will grow to around 25. Algebraically derived
relationships for computational convenience will double or
triple this
number. The key to the utility of the list will be a clear
statement of
what each formula does and where it is most useful for our
purposes. As
a final comment, there are statistical formulas with value to us
such as
arithmetic and perhaps geometric means, standard deviation,
standard
error of the estimate, median, etc. They will be individually
discussed
in the e-mails at the appropriate times and included in total in
the
book on dendromorphometry.
Bob
Robert T. Leverett
Cofounder, Eastern Native Tree Society
|
| Back
to Beth |
dbhg-@comcast.net |
| Apr
25, 2007 13:30 PDT |
Beth,
One of our ENTS objectives is to
create a cookbook recipe of formulas for tree measuring with
very explicit instructions on how to best apply each formula in
the field. Ed Frank is particularly good at writing crystal
clear recipes for applying our methods. Hopefully, Ed will be
heavily involved in that phase.
In terms of what gets posted, some
of the e-mails will be the conceptual stuff and some of it more
practical. Please don't hesitate to ask questions on how to
apply the formulas in real situations. There is no bad question.
Bob
-------------- Original message --------------
From: beth koebel
| |
Bob,
I do read your math heavy emails, although I have to
admit most of it
goes right over my head. I will someday have to sit down
and fiddle with
them to try to understand what you talking about. It
took me no time at
all to understand formula #1 after I looked in my copy
of "Pocket Ref"
(my "little black book"). I guess I'm more of
"here's the formula and
how you use it" then I get it type person.
Beth
|
|
| #15
with reference to #16 |
Robert
Leverett |
| May
02, 2007 10:25 PDT |
ENTS,
Formula #15 for our list of high
use (or potentially high use)
formulas is the differential for height using the tangent
method. For
investigating the magnitude of height errors, the tangent
differential
formula has the same utility as the differential for height
formula
based on the sine method. The height differential formula for
tangent
is:
dH = [D/(cos(A))^2] *dA + tan(A)*dD,
where D = baseline distance, A = angle, and dA and dD are the
differentials corresponding to measurement errors in angle and
distance.
Recalling the corresponding
differential for sine is:
dH = L*cos(A)*dA + sin(A)*dL,
where L = hypotenuse distance, A = angle, dA and dL are the
differentials.
How do errors for the two height
measuring methods compare using
the differentials to calculate error? The examples below show
dHt, the
height differential for the tangent process, and dHs for the
sine
process. Distances of 100 and 150 feet and angles of 30 and 60
degrees
are used in these examples.
D 100.0 150.0
L 115.5 300.0
A 30.0 60.0
dA 1.0 1.0
dL 1.0 1.0
dHt 2.95 12.20
dHs 2.25 3.48
At relatively low angles and
short baselines, the differentials
for an error of 1 degree and a distance of 1 foot are almost
equal for
the sine and tangent methods. But the differentials diverge as
the angle
gets steeper and/or the distance gets longer. At 40 degrees and
a
224-foot baseline, an angle and distance needed to produce the
height of
a Boogerman Pine type tree, the errors are 4.55 for sine and
7.50 feet
for tangent. At very steep angles the error becomes even more
pronounced
for the tangent method. For example, suppose the Boogerman Pine
were
arrow straight with a top visible from 50 feet out. If the top
is
directly over the base and our eye is level with teh base of the
tree,
the top would be at 75 degrees. If this seems unrealistic, just
imagine
that broken branches provide us with a window of visibility to
the top.
Under this configuration, a 1-degree angle error and no distance
error
would produce a height error of 13.03 feet for the tangent
method. A
1-foot distance error without angle error would add another 3.73
feet.
Thus, the maximum error for the combination of the 1-foot
distance error
and 1-degree angle error is 16.76 feet - nothing to blow off. By
contrast, the corresponding maximum height error for the sine
method
under the same configuration is a very modest 1.84 feet. This
should be
enough to catch the attention of even the most
lethargic-thinking tree
measurer.
Turning our attention from
combined errors to just distance, the
maximum error that can ever be made in the height dimension for
a
hypotenuse error of 1 foot IS 1 foot and that error is made
shooting
straight up. So the height error for a 1-foot error in
hypotenuse varies
from 0 to 1 foot as the angle goes from 0 to 90 degrees. There
is no
corresponding situation for the tangent method. The tangent
value is
infinite at 90 degrees. However, assuming that 75 degrees is the
highest
angle for which the tangent method can reasonably be applied to
a tree
like the Boogerman Pine, then the height error for a 1-foot
baseline
error is 3.73 feet as shown previously. The corresponding error
for the
sine method at 75 degrees is 0.97 feet. The difference in errors
for the
two methods based on distance error alone is 2.76 feet.
Obviously angle
error can be the big problem with the tangent method.
A more realistic scenario
for a Boogerman-sized tree is an angle
of 40 degrees – assuming the Boogerman’s height to be around
188 feet
tall by now. With both a 1-degree angle error and a 1-foot
distance
error, the sine method results in a height error of 4.55 feet
and 7.50
feet for the tangent method. The difference of 2.95 feet
suggests that
around 3 feet is the largest likely difference in the height
error made
attributable to the method of measurement, i.e. sine versus
tangent when
the angle error is 1 degree and the distance error is 1 foot
where the
angles aren’t too steep. Note that the distance error is made
in the
baseline for the tangent method and in the hypotenuse for the
sine
method.
So now we have 15 formulas. I
should add Lee Frelich’s crown volume
formula as #16. We’ll probably identify 3 or 4 more before
we’re done to
come out with about 20 highly useful ENTS formulas that deal
directly
with tree dimensions. Adding formulas for Rucker indexing,
champion tree
programs, etc. should put us at around 25. That sounds like a
lot, but
it really isn’t.
Bob
Robert T. Leverett
Cofounder, Eastern Native Tree Society
|
|