Expanded
tangent treatment 
Robert
Leverett 
Feb
06, 2007 05:45 PST 
ENTS,
What follows below is a more complete
treatment than was previously
given for measuring the height of a crownpoint above eye level
using a
clinometer and a baseline between two measurement points aligned
with
the crownpoint. The procedure and formulas that will be
presented are
not substitutes for sinebased measuring, but the procedure can
fill the
accuracy gap resulting from the typical use of a clinometer, at
least
for a limited number of situations. It is presented in the
interest of
completeness. The methods will be shown with ample diagrams at
the Cook
Forest Rendezvous in April.
First a quick review. If the measurer is
able to see top of the tree
and the top is directly over the base, then a baseline from eye
to trunk
and the slope % to the crown top is all one needs to compute
height
above eye level. The slope percent converted to the equivalent
decimal
value times the level baseline distance from eye to trunk gives
the
height of the tree above eye level. This is the standard
clinometerbaseline procedure.
Now what if the crownpoint is not
directly over the base? What can
be done to get an accurate height measurement if one does not
know where
a plumb line from the crownpoint to the ground would fall?
Well, there
is crownpoint crosstriangulation, but that process is
difficult to
implement without two long tapes, an assistant, and continuous
visibility of the crownpoint being measured. However, there is
another
procedure that one can apply.
The measurer positions himself/herself at a
spot where the
crownpoint is visible and shoots the angle to the target (or
percent
slope). The measurer then moves back to a second vantage point
and takes
a second reading with the clinometer. The height of the
crownspot above
the eye position at the first vantage points is calculated by
using one
of three formulas.
DEFINTIONS:
d5 = straight line distance between positions of the eye at
first and
second vantage points. This is the baseline. Note that it does
not go
from measurer to the trunk, which is the traditional baseline,
but from
measurer’s first position to measurer’s second position.
a1 = angle to crownpoint at closer vantage point
a2 = angle to crownpoint at more distant vantage point
a3 = angle between eye positions at the two vantage points
(ideally this
is zero)
FORMULAS FOR THREE SCENARIOS:
(1). Baseline between two vantage points is level (a3 = 0)
h = [(d5)tan(a1)tan(a2)] / [tan(a1)  tan(a2)]
(2). Baseline is not level, more distant point is at higher
elevation
h = [(d5)tan(a1){sin(a3)+cos(a3)tan(a2)}] / [tan(a1)tan(a2)]
(3). Baseline is not level, more distant point is at lower
elevation
h = [(d5)tan(a1){sin(a3)cos(a3)tan(a2)}] / [tan(a2)tan(a1)]
A combination of brackets, braces, and
parentheses are employed to
make the formulas clearer.
It's apparent that cases (2) and (3)
lead to awkward calculations. I
doubt many people will want to use these formulas. However, case
(1) is
more straightforward, and again, please note that the distance
from the
measurer's position to the tree (characteristic baseline) does
not have
to be determined. This may come as a surprise. Also note that
the
crownpoint, eye position #1, and eye position #2 all must lie
in the
same vertical plane. The tangent of angles a1, a2, and a3 can be
determined directly from the right scale of a Suunto clinometer
with a
degrees and percent slope scale. You simply divide the percent
slope
read from the right scale by 100 to get the tangent of the
angle. This
new procedure could be useful when the measurer cannot reach the
trunk
of the tree (across water, surrounded by briars, poison ivy,
etc.).
DISCUSSION ABOUT THE PROCEDURE:
I highly doubt that the above procedure
is going to take the
measuring world by storm. Those who have a laser rangefinder,
clinometer, and scientific calculator can apply the much
superior sine
topsine bottom method. Those who don’t need high accuracy,
but do need
great efficiency in the forest, will likely not employ a
technique that
is calculationintensive. However, for new Ents operating on a
razor
thin budget, saving their pennies for the needed equipment, may
find
this procedure of help.
Bob
Robert T. Leverett
Cofounder, Eastern Native Tree Society

