If you don't find an easy way to solve the perpendicular fit, rotate all your bearings so the (jagged) line is nearly parallel with the x axis and use the common formulas. The difference in results will be negligible.

Perhaps I could use that with your data and help us all learn something new.

Excellent!

Please note that I want to check the R/W width (60') between the platted W. side lots (150 - 154) and the

unplatted E. side lots (parcels 00042 - 00045) along N. Goldwyn Ave.

I only see 7 points in your txt file (.csv file under another name) and only three of them fall near one possible straight line and three near another possible straight line. Point 251 is far from either. The lines, if I have correctly identified them, make a considerable angle far from being parallel, so the distance between them changes by more than 50 units (feet?).

I don't think this is a problem for statistics.

I presume you have located property monuments and want to use a best fit line calculation? Why? Why have software determine the r/w line? To call all points “offline, which is a best fit, is reprehensible. Make and use your own calculation. Some points will be off by a bit and they shouldn’t be considered.

Do you place angle points in the ROW line at every recovered monument on your records of survey?

I'm picturing the best fit line pivoting on the Y-intercept and pointing due E. The points below the line are dangling there due to gravity. The point above the line, well, I don't know how to describe that one.

I'm sorry that my leader arrows are confusing (points 251, 252, and 254). The arrows are pointing to the descriptions of those points. I was trying to avoid crowding the plat. All points on the W. side of N. Goldwyn Ave. fall close to the R/W line. The csv file purposely contains only points 248 - 254.

I think, based upon the first post, I can determine the objective. However, upon seeing the drawing and the point annotations, I think, as a surveying objective, it would be, subject to argument, that a correlation be determined between positions determined as a function of plat dimensions and positions determined from field measurements. In other words, response to "what is the difference from plat determined positions to your measured positions?" is the objective. It is not just fitting measured points to a line but, fitting them to plat positions.

Because, in most jurisdictions, right-of-way dimensions are what they are called to be on the plat, they generally prevail over monument positions. In completing a two-dimensional conformal transformation, it is possible to show the difference or correlation in positions between monuments on the ground and right-of-way positions determined from plat dimensions.

A two-dimensional conformal coordinate transformation solution can be linearized. As such, the solution is less complex or intensive than that for determining a line that minimizes the perpendicular distances from it to a set of points.

Paul Wolf's book "Adjustment Computations: (Practical Least Squares for Surveyors), 2nd ed, ©1980" presents the solution in Chapter 16, "Two-Dimensional Conformal Coordinate Transformations." Other similar discussions and publications can be found. However, Wolf's is specifically presented for surveyors.

Wolf's discussions are repeated in:

Appendix B, Coordinate Transformations in "Elements of Photogrammetry," ©1974

and

Section 17.2, Chapter 17 "Adjustment Computations, Statistics and Least Square in Surveying and GIS," ©1997.

The topic is also addressed in:

Section 10.6 Chapter 10, Mikhail & Gracie "Analysis and Adjustment of Survey Measurements," ©1981

It is not just fitting measured points to a line but, fitting them to plat positions.

I'm not concerned with plat positions. This is not a boundary survey. My only concern is R/W widths as per the plat. Thank you for responding. I appreciate it!

I plotted the points in Geogebra, assuming that the smaller number in each pair is the Easting and the larger is the Northing. Three of the four points on the east side are nearly collinear while the three on the west side don't line up as well and have a different azimuth. Point 251 seems unrelated.

It's hard to guess what assumptions and math lie behind the Carlson numbers. Some kind of line of best fit can always be computed from a set of coordinated points, but it may not have a meaningful physical interpretation.

From my point of view and if I understand correctly, you have a really cool concept here. Find coordinates for points on both sides of the street, compute two lines of best fit, and see how well the math fits. To be sure, there are caveats, but it's a nice idea.

The only input is the coordinates, so they have to actually represent what they say they represent. Carry on and perfect it; it's likely to be valuable.

Thanks for your effort! Seems I provided a bogus coordinate file. Please see the correct attached file.

If the "only concern" is right-of-way width, that information is on the plat. The plat right-of-way dimensions, not the monuments, prevail. Unless it is desired to know or determine right-of-way location on the ground, why is there a need for monument positional measurment. -- Perhaps, I don't understand the problem!

This is from the bare bones FooPlot grapher.

The regression equation from https://byjus.com/linear-regression-calculator/ for these points is

y =

The x-coefficient is almost zero which tells us that the regression line is almost horizontal (due east?) which may be what you're looking for. Seven points is a precious few, but it is indicative, perhaps. There's no check on the right-of-way width, though, and having only two points on the east side diminishes the value of two separate regressions. Also, the x-coefficients would be very large as the lines approach vertical.

Perhaps treating it like a network and doing a least-squares network adjustment might give you better information. All of the points could be included and the computed measurements might be more meaningful.

Thanks for sharing this. It's a wonderful exercise for math applied to surveying.

@field-dog I find it a bit confusing to say you are holding the endpoints of the line segment, and at the same time, list offsets for them. It would seem to me that if you hold the endpoints, the line segment is fully determined. What you could then calculate would be the offsets of the intermediate points, which would give a measure of how well or poorly the intermediate points were set, or resisted disturbance.