| Similar
Triangles |
Robert
Leverett |
| Feb
07, 2007 08:10 PST |
ENTS,
The geometrical concept of similar
triangles provides a powerful
tool for tree measurers. However, as with the percent slope
method, it
is often misapplied to tree height. But, used correctly, the
method
gives us one more tool to work with. Let's look at the concept
of
similar triangles.
Similar triangles are triangles
that have the same overall shape.
One is just a blown-up or reduced version of another. Because
similar
triangles have the same shape, corresponding angles of similar
triangles
are equal and corresponding sides are proportional in length. It
is the
condition of proportionality of side length that allows us to
compute
tree height. But how does the process work?
Assume we have two triangles that
are similar. Triangle ABC is the
larger and abc the smaller where the lengths of the sides are
designated
by A, B, and C for the larger and a, b, and c for the smaller. The
condition of proportionality leads to the following
relationships:
A/B = a/b, A/C = a/c, B/C = b/c.
These are all algebraic
expressions and can be manipulated
algebraically to get equivalent forms such as B/A = b/a, B/b =
A/a, and
b/B = a/A, etc. Now suppose that we form a big triangle ABC such
that A
is the tree's height and B is the baseline from the measurer to
the
trunk. Now, if we can form a smaller similar triangle abc where
we can
measure sides a, b, and c, we can measure the baseline to the
tree B and
can compute the tree's height by using the A/B = a/b
proportionality
relationship. Algebraically rearranging, we get A = B(a/b). This
last
formula is what often accompanies diagrams showing how to
measure tree
height using similar triangles. If side A is vertical and the
two
triangles are truly similar, then the process works. However,
what
happens when side A is not vertical (the line from the
crown-point to
the base is not vertical)? Then the process does not work and
that will
be the case when the crown-point is not directly over the base
of the
tree. Sound familiar?
Similar triangles can also be used
very productively for
determining crown-point offset. However, that determination
requires a
mult-step process that will be explained with diagrams at the
April ENTS
event at Cook Forest SP. In fact, all the material in these
e-mails will
be brought together in what I hope will be our first crack in
producing
an ENTS Guide to Dendromorphometry. BTW, Dendromorphometry was a
term
created by Gary Beluzo, partly in jest and partly in seriousness
to
describe the energy with we Ents expend pursuing our passion.
Gary, is
of course, a first class dendromorphometrist. Gary once made up
certificates for John Knuerr and myself. However, the idea of
formalizing Dendromorphometry as an official ENTS discipline has
merit.
ENTS could issue certificates to bonafide dendromorphometrists -
both in
fun and in all seriousness.
Bob
Robert T. Leverett
Cofounder, Eastern Native Tree Society
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