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 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. 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 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) 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 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.). 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 top-sine 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 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. Bob Robert T. Leverett Cofounder, Eastern Native Tree Society