Commandments of Surveying

The Ten Commandments of Elementary Land Surveying

"As the men started on their way to map out the land, Joshua instructed them, "Go and make a survey of the land and write a description of it. Then return to me, and I will cast lots for you here at Shiloh in the presence of the Lord." [Joshua 18:8, NIV]

(This is written primarily for US readers. I do not know what of it, if anything, is applicable in other countries. Sorry.)

I.

Land Surveys in the United States are of two types: in the original colonies and in Hawaii, Kentucky, Maine, Tennessee, Texas, Vermont, and West Virginia, and some other smaller areas, the states made the original land grants based on the traditional system of "metes and bounds" (meaning points and edges), based on references, distances and compass bearings (and traditionally, such things as oak trees, stream beds and piles of rocks; what an earthquake or flood can do to the older system doesn't bear think on). In the other states, the federal government made the original grants, and for the purpose undertook a general survey of these areas beginning in the early nineteenth century. This system of "Federal Townships and Ranges" defines mostly rectangular plots of land and ignores state and county boundaries, and provides the reference points for finer detail. More on metes and bounds below.

II.

The Federal System consists of a series of master Baselines and Meridians which run respectively, east-west and north-south. The land is divided up into squares 6 miles on each side, called a township. Each township has a unique grid number of the form TndRme, where n and m are numbers and d and e are directions, either N or S for d, and E or W for e. So, township T1NR3W refers to the township located in the first 6 miles north and the third 6 miles west of the closest baseline and meridian, which are also identified in a complete description. Each township is divided into 36 mile-square "sections", numbered in serpentine fashion from the northeasternmost to the southwestern. There is a system of nomenclature for subdividing sections into rectangular pieces. See http://www.outfitters.com/genealogy/land/twprangemap.html for graphics of the way this works, and more details.

III.

Obviously, the world isn't flat, even just the relatively small area covered by the United States. Tiling the entire United States in rectangles just doesn't work. That is why there is a system of baselines and meridians, rather than just one each. The differences in the shrinkage of the distance between meridians as we proceed north is taken care of by allowing the six-mile grid system to "slip" along the baselines, thus breaking up the US into a series of flat facets. The survey is then performed on the facets, and then projected onto the real globe. The federal system also admits to occasional errors in the land survey, making some lines less than straight. Well, nothing's perfect.

IV.

One of these two systems establishes reference points from which local land surveying is done. The local surveying, which establishes property lines within platted residential areas, is a metes and bounds system. That means that it starts at a reference point and then defines a property line based on distances and bearings (and a few other things, as we'll see below). Given the list of bearings and distances, the survey should, ideally, result in closure, so that the property is bounded on all sides and that all similar properties essentially account for all the land within the platted area. When a survey is done, 1/2" steel pins are driven into the ground at corner points, and these pins effectively define ownership of the land unless a newer survey moves them.

V.

The first problem with metes and bounds is that not all edges are straight. As far as I have been able to tell, all curved lines on surveys are circular arcs (no other conics or more exotic curves are used). A curved line is defined, not with a bearing and distance, but rather with a radius and a distance, the radius being the radius of the circle that the arc is a part of. The distance is the curved distance that the arc covers. There is a formula to relate the radius "r" and the distance along the curve "d" to the angle "a" of the circle that the arc sweeps out ("subtends"); it is

a = (360 * d) / (2 * pi * r)

which returns a value in degrees and fractions thereof.

Notice that this definition of circular arcs is rather flakey. In particular, if two such arcs meet, their point of meeting can be either of two points in general. If there are three curves in a survey, there are infinitely many solutions, as the curve end-points are not well defined. The piece of the puzzle that allows this to work is that the curves have to be defined such that the curves are smooth. Those who have taken Calculus know about a rigorous definition of smoothness where piecewise curves meet. Since all curves are circles, the rules for survey geometry are understandable to mere mortals, and there is but two:

A pair of straight lines can meet at any angle
If one or both is a curve, then the tangent line of the curve(s) at the meet point can only be at right angles to the straight line/other tangent, or they can meet at zero angle (i.e., they are the same line).

This constraint on curves makes it possible to locate the arc's center point, and thus make it possible to geometrically plot the curves.

VI.

Let's do an example. A user has sent me a survey of his property (Thanks, Rift). It consists of three straight line segments and three curves. Written out, the description of the land is: "Starting at the base point (I'll assume that's the point in the southwest corner), proceed along these bearings (b) and distances (L), or along curves with radius (R) distance (L):

B=N 36° 26' 42" E, L=139.48 (read the angle as a little more than 36 degrees East of due North)
R=50.00', L=13.16'
B=S 68° 38' 0", L=24.82'
R=25.00', L=36.36'
R=620.00', L=83.12'
B=N 82° 59' 28" W, L=117.78'

To get the data organized, let's build a table with everything we need:

Edge Number	Distance, feet	Azimuth, degrees	Radius, feet	Subtended Angle, degrees	Center side
1	139.48	36.45
2	13.16		50.00	20.44	outside
3	24.82	111.37
4	36.36		25.00	83.33	inside
5	83.12		620.00	7.68	outside
6	117.78	277.01

VII.

I used the Punch! Surveyor ("Site Planner") to layout the first and last edges. It did a great job, but it does not handle curves, so we have to proceed with the more general drawing features of the detail tab. (one could also derive the chord distance and chord angle - the "chord" is the straight distance from end-to-end of a curved arc, and then treat that as a straight line which can be used in the Surveyor, as long as one is cognizant of the distortion this produces in the completed survey. See article VIII below.) The center of the first curve's circle has to be located along the extension of edge 1, 50.00' beyond the meet point (beyond because it is outside). I copied the 50' extension after it was drawn, and rotated that by a, and moved the points together at the circle center; that located the end point of the curve. (The green curve was added using the circular arc tool, which is rather approximate, so the measurements don't depend upon it. It is only there for appearances.) The remainder of the curve centers were similarly constructed, and, surprise, the ends came together within a few inches. I couldn't construct the green curve for the long radius circle because a circle of that size can't be drawn on the Punch! drawing surface, so it was approximated with three straight line segments. The final result is shown here:

VIII.

There are two other formulas which should be added to this discussion; one that relates the properties of circles above to the chord, which is the straight line between the endpoints of the arc. First the chord length:

c = 2r * sin (a / 2), and its inverse: a = 2 arcsin (c / 2r)

and the angle (b) I'm calling the chord angle, which is the angle that the chordmakes to the tangent:

b = a / 2

which may be added or subtracted from the bearing leading to the meet,l depending on the way the curve breaks from the incoming bearing.

If we were to extend the table above with these two figures, it would look like this, computing edges in a clockwise fashion indicated by the edge numbers:

Edge Number	Distance, feet	Azimuth, degrees	Radius, feet	Subtended angle, degrees	Center side	Chord length	Chord bearing
1	139.48	36.45				139.48	36.45
2	13.16		50.00	20.44	outside	13.12	116.23
3	24.82	111.37				24.82	111.37
4	36.36		25.00	83.33	inside	33.20	159.67
5	83.12		620.00	7.68	outside	83.04	190.73
6	117.78	277.01				117.78	277.01

The only real reason to go through this additional pain is so that the Punch! Surveyor can be used all the way through, with the understanding that the resulting lines are only approximations when they replace curves. Applying these figures in the Surveyor results in:

IX.

The tools in Punch! are not well suited for this application if curves are involved. That is too bad; perhaps the Surveyor will be upgraded to do this at some point in the future.

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This page last updated on Sun May 14 2006
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