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
