A 471Kb PDF of this article as it appeared in the magazine—complete with images—is available by clicking HERE
Dr. Elgin, The American Surveyor Staff, and I collectively strive to educate our peers and readers of this publication. I have continually enjoyed Dr. Elgin’s effort to advance knowledge through problem solving. The HP 35s is a true problem solver and I find great honor in demonstrating how to solve one of Elgin’s enigmas with this tool. The goal is for everyone to appreciate the mechanics of problem solving and instill the necessary logic to tackle larger problems. There is no digital witchcraft or ghost in the machine here, nope. Just logic, brain cells, and two CR2032 batteries. You may find several approaches to solving any given problem. The more the merrier, I say! Let’s refer to TAS October 2015 issue. Dr. E’s ultimate request is to "Compute the radii of the replacement curves" so that a 100.00 foot tangent separates them. This problem is all about the tangents. First, let’s organize our given information.
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There is something we need to understand along with this simple curve data. With reverse curves the tangent direction out of C1 is the same as the tangent direction into C2, or a "straight line" to us hill folk. Consequently the distance between P.I.’s in a reverse curve is simply the sum of their tangent distances. That’s very convenient because we can solve both tangent values by simply multiplying the TAN of one-half delta by the radius.
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So the distance between the P.I.’s on a straight line could be expressed verbally as "the tangent of half of one-hundred and twenty degrees fifty-seven minutes times three-hundred plus the tangent of half of one-hundred and four degrees thirty-seven minutes times one-hundred and sixty."
Or
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In our case the solution equals 736.79 feet between points. Dr. E. wants 100.00 feet of tangent to separate P.T. #1 and P.C. #2 and we shall deliver. "Seven-hundred thirty six point seventy-nine feet minus one-hundred point oh-oh feet equals six-hundred thirtysix point seventy-nine feet". Done!
736.79-100=636.79
We have 636.79 feet of tangent to distribute among each side of the 100.00 tangent. Keep in mind that after the dust settles our final tangent values should equal this number. The challenge of this whole problem lies in writing an equation that will express our given ‘s, equal radii, and maintain 100.00 feet of tangential separation. Hmmmmmm…
Let’s start by recognizing the facts. Our ‘s are fixed and our unknown radii values are constrained by the condition of being equal. That leaves us with elastic tangent values, right? So another table:
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Ultimately we are solving for "R" which according to Dr. E must be the same value in both curves. So, if we can form a statement from C1 that equals "R" and the same statement applied to C2 also equals "R" then doesn’t it stand to reason that both statements are equal? Yes, it’s true and we can form a larger statement by substituting smaller statements into singular variables. That’s handy because the HP 35s EQN Library and solver also will do just that. Let’s jot down equations that express some common elements between each curve:
We’ve already figured out distance between P.I.’s minus Dr. E’s requested hundred feet.
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Jason Foose is the County Surveyor of Mohave County Arizona. He originally hails from The Connecticut Western Reserve Township 3, Range XIV West of Ellicott’s Line Surveyed in 1785 but now resides in Township 21 North, Range 17 West of the Gila & Salt River Base Line and Meridian.
Sidebar:
Equations & Solver
The HP SOLVER and EQN Library are very powerful tools. Simply queue up any equation and BRS EQN initiates the SOLVER. You will be prompted to select a variable to solve for and input the remaining variables. You can solve for any variable residing within the equation regardless of its location. Variables are stored so you may resolve a series of progressive equations and carry variable solutions, or better yet write the equation itself incorporating the sub-equations for the variables. For example Y=mX+b represents how many "Y’s" are equal to a certain number of "X’s" where X progresses from a known point (b) at a certain rate (m). "m" (slope) is equal to (the differences in your sample points). So instead of hand computing "m" by itself, why not just substitute "m’s" equation in place of the "m" variable? Get it?
A 471Kb PDF of this article as it appeared in the magazine—complete with images—is available by clicking HERE