"Biltmore" Mathematics

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Problem Description

Here's a program to calculate the important layout parameters for constructing a Biltmore Stick to visually estimate tree diameters . 

Background & Techniques

The Biltmore Stick was first developed (or designed or invented) in the early 1900's to help in a large forestation project on the Biltmore Estate in North Carolina.  It uses a variation on the principle of lining up the eye with two other objects to determine a straight line.  Student pilots line up their eye, the nose of the airplane and the end of the runway to guide their plane down safely.  Farmers set their fence posts in a straight line by sighting back to two previously set posts (when you only see a single post, your eye is over the point for the next posthole).  Shooters need to line up 4 points, the eye, the rear sight, the front sight and the intended target.

With a Biltmore Stick, the stick is held at arm's length and at right angles to the tree with the marked zero point on the stick in line between the eye and one edge of the tree.  The reading marked on the stick which is in line with the eye and the other edge of the tree, is the diameter.  This program addresses how to place the diameter markings on the stick.
 

The Math

Here's the above diagram with some points labeled:

DAB represents the Biltmore stick held at right angles to the tree with end (or zero point) D lined up between the eye at E and tangent to the edge of the tree at F. The stick is held at a specified length, L, from the eye location. We want to determine the distance, s, from D to B, the point on the stick directly between the eye at E and the right edge tangent at C. We can then label point B on the stick with the diameter of  the tree, d. Here's the derivation:

Triangles EOC and EBA are similar (they are both right triangles and share angle AEB).  Therefore the ratios of the shorter to longer non-hypotenuse sides are equal: AB/AE = OC/EC or AB = OC*AB/EC.

But AB = s/2, AE = L, OC = d/2, and EC = sqrt(EAO)2 - OC2) = sqrt((L+d/2)2 - (d/2)2 )),

Expanding and reducing: EC = sqrt(L2+ 2Ld/2 + d2/4 - d2/4) = sqrt(L(L + d))

So we can write AB = s/2 = AE*OC/EC = L(d/2) / sqrt(L(L + d))

And doubling both sides: s=Ld / sqrt(L(L + d)), one common measuring formula ;

Squaring both sides: s2 = L2 d2 / ( L(L + d))

and s = sqrt( (Ld2)/(L + d) ), an equivalent formula for creating a Biltmore stick.

Running/Exploring the Program 

bullet Download source
bullet Download  executable

Suggestions for Further Explorations

 The commercial sticks usually also include a Hypsometer scale for estimating tree heights based on the same principles described here but with the stick held vertically (and from much further back from the tree than arm's length! ).  Adding a page to calculate that scale should be a simple addition.

 

Original Date: December 6, 2010 

Modified: July 29, 2017

 

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