The HP 35s Calculator—A Field Surveyor’s Companion: Part 6—Curve Traverse

A 363Kb PDF of this article as it appeared in the magazine—complete with images—is available by clicking HERE

This Month’s Program
This program is a curve traverse routine based upon the traditional methods of laying out a curve with a transit and tape. I assure the users of radial layout equipment and GPS that using this program is 100% compatible with all current land survey measuring systems, as well as any electronic survey data collector/total station developed after January 1, 1959. Those who understand why may take a moment to bask in a jocular glow of professional enlightenment and those finding comfort in my assurances may wish to attend my "How to make a jillion dollars by giving me $99.99+shipping and handling" seminar at the Thunderbird Conference Center in Granger Township on the 23rd of this month.

Prior to total stations, electronic data collection, and GPS, the basic method for laying out a curve was to physically occupy the p.c. and sight a point on the back tangent, or the p.i, or the p.t. The "Delta" or central angle was computed for the each of the desired points on the curve along with the chord distance from the occupied point (p.c.). To solve for Delta, I simply divide the desired arc length by the radius. This produces the delta in radians. The HP 35s keystrokes BRS .9. converts radians to decimal degrees. In the days of yore this conversion could have been done by hand through the formula Decimal Degrees=Radians x 180/ or simply pulled from a table. The deflection angle from the tangent line is equal to one-half of Delta for any given curve. This establishes our "stakeout" angle or direction from the instrument. The chord distance is solved by the formula "two times the radius times the sine of half Delta". This establishes our "stakeout" distance from the occupied point to the point on the curve.

So, all of this collectively applied yields the following process:
1. Set on p.c. and sight p.i. with 0°.
2. Make a data table of Delta, Half Delta, Chord Distance, Arc, and Radius for each point relative to the p.c.
3. Turn Angle Right or Left to equal the Half Delta for a given point.
4. Set the point at the Chord distance.
5. Repeat for all inter-visible points.

The program functions with and through this logic. It is set up to return to the AZ TRAVERSE program after completion. This is a nice segue from leaving a full curve and going through the tangent out. This can be modified to the user’s preference at line D034. Generally your "tangent in" is going to be the same as the previous bearing leading up to the P.C. So at the "TAN AZ FWD" prompt you can simply manipulate the bearing into the 360° azimuth using the active stack under the prompt. The results are applied by the .R/S. key. The ability to crank numbers with the stack during an open prompt is one of my favorite features of the 35s. Please do not hesitate to send any comments, concerns, questions, or criticism to

The Program

Curve Retracement Comment
The examples herein are being offered to demonstrate a generalized method of curve layout. Understanding the fundamentals of curve layout is a defining circumstance of our legal function as retracement surveyors. Suppose we are employed to identify the boundaries of a curvilinear tract of land originally subdivided prior to recording laws, electronic equipment, or formal recognition of monumentation standards. Utilizing GPS to collect a positional attribute provides only two bits of info for analysis.

1. An independent position of the evidence itself; and,
2. A string of text or photo to demonstrate the physical characteristics of the remaining evidence.

Negligently dispatching a field crew to simply "collect" the positions of boundary corners narrows the Surveyor’s analysis to the singular function of comparing a collection of independent data points against platted geometry under the assumption that the plat exactly matched the work in the field. This method begins to discolor in a shade of fraud as the Land Surveyor is presumed to know that some imprecision and tolerance is appropriately applied between the original ground markings and the original plat calls. The unprofessional tendency to simply perform a comparative analysis of precise GPS positions against a geometric depiction of a lineal survey offers but one analytical element. That being the difference between reported relationships on the plat vs. indirect positional relationships observed by the GPS. This unprofessional approach provides zero accountability for the allowable tolerances and imprecisions derived on the ground at the time of the original survey.

A true retracement survey would better serve its purpose through application of the original techniques expected from the original surveyor. For example physically occupying a pipe at the P.C. with a modern total station and observing the deflection angles and chord distances to the points on the curve may reveal the following:
1. The deltas were only calculated to a precision consistent with the original transit.
2. All curve points look pretty good from the P.C. and landed where the original surveyor said they would, but none fall precisely on my computed overall curve.
3. An object obstructed the line of sight along a calculated deflection angle so the point was "eased in" around the limited sight line.
4. All the points fit the deflections very well but the distances were loose.
5. Acceptable limitations of precision related to the use of slide rules and tables.

None of these conditions listed above demonstrate any legal concern promulgating the rejection of a monument placed in its original position. However, every one of the conditions reinforces, if not proves, the positive identification of original evidence or the perpetuation thereof.

A purely geometric comparison can only produce the statistical insecurity of nonconformance with the plat. Conversely, practicing the instruction "to follow in the footsteps" provides professional insight applicable to the evaluation of bona fide evidence. The ability to understand and replicate the methods employed by the original surveyor imparts the retracement surveyor with the opportunity to assess the original imprecision between the original points set on the ground vs. the reported calls of the original plat. This essential aspect of boundary analysis nourishes our acceptance and perpetuation of bona fide evidence. Whereas, the statistical comparison of positions only establishes an artificial rejection parameter without providing any positive attribute to support the position.

Logic employed in the Curve Traverse Program follows in the tradition of transit/ tape methods. You’ll notice that the term "Radius Point" is not mentioned nor is necessary to the function of laying out a curve. The "radius point" was normally not a part of the field operation and therefore is mostly irrelevant to successful retracement and perpetuation of original evidence.

Jason Foose is the County Surveyor of Mohave County Arizona. He has been licensed for  441,504,000 seconds…no wait, 441,504,001 seconds…no wait, 441,504,002 seconds…

A 363Kb PDF of this article as it appeared in the magazine—complete with images—is available by clicking HERE

About the Author

Jason Foose, PS

Jason Foose 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. He is also the Managing Editor of the magazine.