Expanded tangent treatment   Robert Leverett
  Feb 06, 2007 05:45 PST 


   What follows below is a more complete treatment than was previously
given for measuring the height of a crown-point above eye level using a
clinometer and a baseline between two measurement points aligned with
the crown-point. The procedure and formulas that will be presented are
not substitutes for sine-based 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
clinometer-baseline procedure.

    Now what if the crown-point 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 crown-point to the ground would fall? Well, there
is crown-point cross-triangulation, but that process is difficult to
implement without two long tapes, an assistant, and continuous
visibility of the crown-point being measured. However, there is another
procedure that one can apply.

   The measurer positions himself/herself at a spot where the
crown-point 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 crown-spot above
the eye position at the first vantage points is calculated by using one
of three formulas.


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
measurerís first position to measurerís second position.

a1 = angle to crown-point at closer vantage point

a2 = angle to crown-point at more distant vantage point

a3 = angle between eye positions at the two vantage points (ideally this
is zero)


(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
crown-point, 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.).


    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
top-sine bottom method. Those who donít need high accuracy, but do need
great efficiency in the forest, will likely not employ a technique that
is calculation-intensive. However, for new Ents operating on a razor
thin budget, saving their pennies for the needed equipment, may find
this procedure of help.


Robert T. Leverett
Cofounder, Eastern Native Tree Society