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
 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

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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