I don't want to start the argument again about which to use or whether it matters.?ÿ I just wanted to compare methods of calculation and thought others might be mildly interested.?ÿ Thanks to @mathteacher for a spreadsheet that helped with some of the calculations.
-------------
Midpoint of Easterly-Westerly line
If setting a lost quarter-section corner on a nearly east-west line, there are several possible methods that give similar, but not identical, results. The results fall into two groups, all within about 0.17 ft of each other (northern US, less at lower latitude), so it isn??t a big deal, but it would be nice to know the theoretical differences.
This is similar to the center of section argument, but lacks the complication of finding an east-west position that may not be particularly near the midpoint.
Let??s work on the ellipsoid to avoid height complications. Pick a theoretical line in the northern part of the US where convergence is larger, near the corner of a UTM zone, with a Lambert SPC.?ÿ?ÿ A point in northern Minnesota meets these criteria.
Assume an east end at round numbers, and take the west end to be 1609.000 m (5278.861 sFt ) at geodetic azimuth 273?§. Conversions were done with Corpscon6.
|
?ÿ |
Lat N 48?§00?? + |
Lon W 95?§30??+ |
SPC 2201 N, meters 269,000+ |
SPC 2201 E, meters 619,000+ |
UTM 15T N, meters 5,319,000+ |
UTM 15T E, meters 310,000+ |
|
East pt |
0? |
0? |
540.593 |
1944.591 |
324.564 |
3517.365 |
|
West pt |
0?? 2.71915? |
1?? 17.51449? |
674.628 |
341.333 |
460.840 |
1914.096 |
|
A1 visual, geodesic |
1.36139? |
38.75696? |
607.611 |
1142.962 |
392.702 |
2715.732 |
|
A2 mean Lambert |
1.36137? |
38.75697? |
607.610 |
1142.962 |
392.701 |
2715.732 |
|
A3 mean TM |
1.36140? |
38.75706? |
607.611 |
1142.960 |
392.702 |
2715.730 |
|
B4 mean Lat-Lon |
1.359575? |
38.757245? |
607.555 |
1142.955 |
392.646 |
2715.724 |
|
B5 const bearing |
1.359720? |
38.757240? |
607.559 |
1142.955 |
392.650 |
2715.725 |
?ÿ
Group A, all within 2 mm:
- Visual midpoint of line. Set up on one end and get azimuth and distance to opposite corner.?ÿ?ÿ Set the point at same forward azimuth and half distance, either by instrument or by using NGS Inverse and Forward.
All three points lie on a geodesic line of the ellipsoid (great circle if on a sphere). From the other end, the azimuth is slightly different, but the midpoint comes out identical. Also equivalent would be to wiggle in to the midpoint so you optically measure 180?§ azimuth difference and equal distances.
Clairaut Constant calculation of a geodesic line by spreadsheet gives the same answer, if done to sufficient accuracy, with attention to the change in earth radius.
- Lambert projection, mean coordinates of the end points. Since we assume the line is near east-west, the distortion will not cause much difference from the other methods in this group.
- Transverse Mercator projection, mean coordinates of the end points.?ÿ?ÿ The distortion (change in scale factor) caused by moving West, perpendicular to the N-S axis of the projection, does not appear to prevent this from giving a good match to the other methods. It only moved the ??midpoint? west 2 mm.
Group B, within 5 mm of each other, and about 52 mm (0.17 ft) south of Group A
- Mean latitude and longitude of the end points. This falls just 4 mm south of the constant bearing method, and may be accurate enough for the purpose.
- Bearing held constant (a loxodrome or rhumb line) from one section corner to the other, marked at the midpoint. A spreadsheet in 1 meter increments may have sufficient precision.
A sufficiently accurate rhumb midpoint value for any approximately E-W line can be found more easily using Inverse and Forward. First calculate the midpoint of a geodetic line starting and ending at the average latitude and actual longitudes, and then the midpoint of a constant latitude line. Find the offset between those and apply it to the geodetic midpoint of the actual section corners to approximate the rhumb line midpoint.
The conclusion is that if you choose to worry about the small difference, you don??t need fancy calculations to find your point. You can use the average of a plane projection for the visual/geodetic method and the average lat-lon to approximate the constant bearing method. You will be within a half-centimeter or a couple hundredths of a foot from theoretical.
?ÿ
I reworked the spreadsheet for the constant bearing line and now it comes out at almost exactly the mean latitude, and 3 mm east of the mean longitude.
?ÿ
Howdy Bill,
Out of curiosity, I ran your "scenario" through my personal "latitudinal arc" software that I wrote many years ago in BASIC. It isn't the most elegant piece of software (by a LONG shot), but has served me well for decades.
My solution for the "mid-point" is;
48?ø00'01.35967"
95?ø30'38.75723"
How does that look with your latest revision?
Loyal
?ÿ
You are 4 mm = 0.013 ft at 318 degrees from my constant bearing midpoint.?ÿ Validation for both of us.
?ÿ
You are 4 mm = 0.013 ft at 318 degrees from my constant bearing midpoint.?ÿ Validation for both of us.
Yeah...I can live with that!
?ÿ
You are 4 mm = 0.013 ft at 318 degrees from my constant bearing midpoint.?ÿ Validation for both of us.
Yeah, validation that you are both probably wrong. 😉
(smartassfontoff)
What constant bearing did you assume for the rhumb line?
When I started to use Autocad (about mid 80's)), I realized that I could draw some township lines in state plane at a latitude using SPC coordinates. So I put together a drawing with township lines at even degrees of latitude that would cover any area I might work in. A line 6 miles long with each end at a latitude say 40 degrees, then one at 41, 42, ect. and placed a coordinate at each 1/2 mile. Then I shifted each coordinate to the latitude and I had a diagram with 6 miles and an easy way to find the offset to geodetic line for that lat. Of course the actual lines on the earth wouldn't be at an even latitude, exactly a mile long or due east or west, but a few 10s of feet per mile, a 1/2 degree of latitude or a few minutes from due east west wouldn't make much difference to the offset at one mile or 1/2 a mile. For the majority of my work .88' in a mile (mid point of two miles), .16' in half a mile and .04' for a quarter is very close. It was a simple way to calculate the location in the field, just use the chart.?ÿ
Of course when I got GPS all that went away and it was simple to just calculate it right there in the field without much effort. Not exactly by the book, but I never had a complaint from the BLM who would sometimes follow me.?ÿ
Yep. The math is just a model, not the earth itself. Like in the loan business; you never see an amortization schedule with a default or a payoff planned after 10 months, but they do happen. The schedule is a model that fits perfectly under certain circumstances.
In the current case, the geodetic line and the rhumb line are two entirely different lines. If you use the azimuth of the geodetic line for the rhumb line, the rhumb line will not go through the defined end point. If you make a rhumb line go through both defined points, its constant azimuth will not be the 273 degrees initial azimuth for the geodetic line and it will be longer than 1609 meters by a little bit. The geodetic line is the shortest one between two defined points, so any other line will be longer.?ÿ
The midpoints of the lines cannot coincide.
It's really an exercise in splitting hairs because the differences are insignificant, but we do need to know the characteristics of the models we use.?ÿ
I have tried numerous methods, most of which come close to each other. In the end, table 11 of the red book gives a sufficient math answer. Arguably it is the best answer as that is how most original surveys were done here...
Get a solar compass and run the line that way.
Or just use software (eg. Carlson, Topsurv..) that can automatically do the calculations for you.?ÿ
Speaking of old methods and straight lines, pages 113 and 114 of this report from 1898 or so give a good description of 10 proxies for straight lines used in the 1891-96 survey of the US-Mexico boundary. There's actually a lot of other good stuff in this publication, including some really cool photographs, but entering 113 for the page in the Google Books viewer gets to the relevant stuff.
There is a paper by Rod Deakin on the Curve of Alignment, the line on an ellipsoid whose points are all co-planer, making it a straight line, that discusses the math that models it. It used to be free and there may still be a free download site somewhere. It's pretty complicated stuff.
Geez, I get more math nerdy every day.
