New Formula

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ryandallas
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New Formula

Post by ryandallas » Tue Jul 28, 2015 9:35 pm

I don't know if anyone's presented this formula. If someone has, I apologize for stepping on his/her roots (he he).

Imagine you're standing on high ground--say, on a mountainside or a bluff. Below you is an alluvial flat, on which a fine tree is growing. You're well above the tree's top; the standard sine method is of no use. The following formula might determine the tree's height:

Ht. = [√C²- (sinG * C)²] - [√F²-(sinH * F)²]

I've posted an explanatory photo below.

The laser shots towards the tree's top (F) and the tree's base (C) suggest two right triangles, which are pictured below. The angles of laser shots C & F are represented by letters G & H, respectively. One can determine all of the triangles' sides by using trig and the Pythagorean Theorem. In the case of the smaller triangle, for example, side D would be the sine of angle H times length F. To determine the length of side E, one would then find the square root of F² - D². Side A could be determined using this same math.

Once all sides have been determined, subtracting the smaller triangle's adjacent side (E) from the larger triangle's adjacent side (A) yields the height of the tree.
Attachments
Tree diagram.doc
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Don
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Re: New Formula

Post by Don » Wed Jul 29, 2015 12:56 am

Ryan
Good to see innovative approaches...these are the kinds of problem solving approaches that define Cadre members.
That said, uhmmm, Cadre members correct me if I'm wrong, but the sine-sine method should work just fine, given angles to top and base, and slope distances to top and to base...it gets a little tricky with the plus-es and minus-es is all.
-Don
Don Bertolette - President/Moderator, WNTS BBS
Restoration Forester (Retired)
Science Center
Grand Canyon National Park

BJCP Apprentice Beer Judge

View my Alaska Big Tree List Webpage at:
http://www.akbigtreelist.org

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dbhguru
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Re: New Formula

Post by dbhguru » Wed Jul 29, 2015 10:37 am

Don,

Yes, you are right, the sine method works whether the object is wholly below eye level, wholly above, or part above and part below, since as we've discussed, the sine method computes the vertical separation of two points in space (with respect to gravity) regardless of the positioning of the points relative to the measurer.

Ryan,

We congratulate you on thinking about these issues. Please don't stop. It shows that your are looking at the underlying mathematical relationships, which is exactly what we want. Below is an illustration of two approaches, plus a simplification of the first.
Screen shot 2015-07-29 at 9.08.17 AM.png
If you have the hypotenuse and angle of the hypotenuse of a right triangle relative to the eye, the lengths of the other two sides of the triangle (opposite and adjacent sides) are automatically determined through the trigonometric equations that we commonly use. The pathagorean relation illustrates the association between its pure form (square of the hypotenuse equals the sums of the squares of the other two sides) and trigonometric substitutions for the other two sides as shown below.
Screen shot 2015-07-29 at 11.18.32 AM.png
Screen shot 2015-07-29 at 11.18.32 AM.png (17.64 KiB) Viewed 1045 times
A point that all Cadre members need to be able to explain to others is that for a point positioned in space, relative to the eye, with direct distance of eye to the point equal to L, and L at an inclination angle A to the eye, the length of the correct baseline for the point is Lcos(A). If it is a tree that we're measuring, the trunk is incidental to the calculation. The trunk is traditionally used to establish a baseline for the crown measurement stemming from our lack of having a device that measures the distance to the target directly. So, a stretched tape to the trunk created a surrogate baseline - with the assumption that the top of the tree being measured was positioned vertically above the end of the trunk baseline. That measuring protocol was repeated down through the decades and became canonical. The introduction of the infrared laser rangefinder has changed the tradition.

Bob
Robert T. Leverett
Co-founder, Native Native Tree Society
Co-founder and President
Friends of Mohawk Trail State Forest
Co-founder, National Cadre

ryandallas
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Joined: Fri Mar 07, 2014 5:23 pm

Re: New Formula

Post by ryandallas » Wed Jul 29, 2015 4:59 pm

Thank you for the helpful feedback, guys.

Don,

I had not seen this formulation of the sine equation before--and it does indeed work beautifully. I was thinking of another equation, I believe.

Bob,

Your equation is the simpler, and thus the superior. The one I came up with is clever, but yours is more practical, more elegant, etc. I will have to read up on it some more! I am above average, but not exceptional, mathematically, so it might take me a while. :/

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Don
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Re: New Formula

Post by Don » Thu Jul 30, 2015 11:24 am

Ryan-
Well done!
I feel I'd be remiss if I didn't point you to other explanations of the Sine Method in:
http://www.americanforests.org/wp-conte ... nes_LR.pdf
pages 44-50. There are some simple algebraic manipulations necessary for dealing with positive and negative values due to the relationship of the observer and object (i.e., being above and/or below trees top/base.
Don Bertolette - President/Moderator, WNTS BBS
Restoration Forester (Retired)
Science Center
Grand Canyon National Park

BJCP Apprentice Beer Judge

View my Alaska Big Tree List Webpage at:
http://www.akbigtreelist.org

ryandallas
Posts: 115
Joined: Fri Mar 07, 2014 5:23 pm

Re: New Formula

Post by ryandallas » Mon Aug 03, 2015 8:41 pm

Thank you, Don! I'll look at it right away.

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