Activity Feed › Discussion Forums › Strictly Surveying › Helmerts mathematics

# Helmerts mathematics

Posted by summerprophet on May 8, 2023 at 2:29 pmHi all,

I am making an excel spreadsheet to do a helmert’s transformation. This basically mimics the old Helmerts program for the HP48.

What I am lacking is the “Set scale to 1” function, which would be valuable. Unfortunately, my knowledge of the mathematics is lacking, and I uncertain how this step is done. Most of the hemerts research i have uncovered involves graphing it in excel and is not specific to surveying.

If anyone is willing to give it a look, I can provide the spreadsheet and the sample data.

mathteacher replied 1 year ago 4 Members · 6 Replies- 6 Replies

How many parameters are you looking to perform 3 or 7 etc.

There are a few different approaches to solving these in general. I have implemented at least 3 different versions of the 2d similarity transformations over the past ~15 years so I might be able to help. Please post or send the Excel file and I will have a look. It’s typically quite straight-forward to force the scale to 1.

Jacob WallJust a 2d transformation.

@summerprophet gotcha.

@jacob-wall

Private message sent.Apparently, the answer to the question of finding Helmert 2d coefficients with a scale factor of 1 was answered by private message. Maybe it’s arcane, but exploring the process yields some valuable insights.

First, the short answer to the question. The two equations that rotate and scale one map projection into another via Helmert 2d transformation are:

where: N’ and E’ are the transformed coordinates,

N

_{0}and E_{0 }are constants determined by analysis,s is the scale factor determined by analysis.

and is the rotation angle determined by analysis.

The typical analysis method is a multivariate linear regression.

Just a little independent algebra yields:

Thus, we calculate the scale factor

*s*and divide the coefficients of N and E, but not the constants, in both of the above equations by that value of*s*. The resulting equations have a scale factor equal to 1.Here are the coordinates of 5 points on an LDP near Pinehurst, NC that are also NGS survey marks. They are identified by their respective PIDs. The LDP is a Lambert conformal projection tangent to the topographical surface at 35° 10’ North and 79° 29’ West, origin at N = 25,000, E = 100,000, and k

_{0}= 1.00001579.LDP

NC SPCS

Point

Northing

Easting

Northing

Easting

DH9398

24,269.961

99,569.154

156,522.732

566,657.950

DH9405

26,643.312

99,991.063

158,893.737

567,090.950

DH9410

27,403.534

102,077.674

159,644.035

569,180.813

DH9408

26,128.143

102,380.720

158,367.415

569,477.819

DH9397

24,066.459

102,671.253

156,304.681

569,758.619

The task is to rotate and translate the LDP into State Plane using a 2d Helmert transformation. Least squares linear regression produces the following equations:

Northing

_{SPCS}= 132,724.475 + 0.99984696*Northing_{LDP}− 0.00470014*Easting_{LDP}Easting

_{spcs}= 466,989.975 + 0.99984695*Easting_{LDP}+ 0.00469968*Northing_{LDP}The coefficients are

*products of a scale factor and a trig function*. The two cosine coefficients should be equal and the two sine coefficients should be equal, and they’re very close. But we’re dealing with squaring numbers with 9 significant digits in Excel. The correlation coefficient is not exactly one in either case, so there’s a millimeter difference between some of the actual SPCS coordinates and their corresponding transformation values.From the equation for Northing

_{spcs}:Our coefficients with s = 1 become:

0.99984696/0.99985800 = 0.99998896 and 0.00470014/0.99985800 = 0.00470081.

These are the cosine and sine, respectively, of the rotation angle. For a quick check of the process, we can compare the rotation angle to the convergence angle on the data sheet for DH9405. Why should these values be so nearly equal?

The LDP coordinates are ground coordinates. Using the coefficients that include s < 1 produces

*state plane coordinates*. The scale factor scales the coordinates from ground to grid. Comparing transformed coordinates to data sheet state plane coordinates checks this claim.Using the coefficients with s = 1 produces

*ground coordinates*. The scale factor leaves the LDP scaling intact. This can be checked by comparing distances computed from LDP coordinates to corresponding distances computed from coordinates transformed with s = 1.Here are the transformations:

Transformed to SPCS, s<1

Transformed to Ground, s=1

Northing

Easting

Northing

Easting

156,522.732

566,657.951

156526.1124

566672.118

158,893.737

567,090.949

158897.4539

567105.179

159,644.035

569,180.814

159647.8587

569195.341

158,367.415

569,477.819

158371.0572

569492.388

156,304.681

569,758.618

156308.0303

569773.227

This is a perfect example in that both sets of coordinates are geodetically determined on the same ellipsoid and by the same map projection. It lets us understand the process, but real-world transformations are likely to be much messier and have poorer correlations. Each has to stand on its own merit.

Here are three useful references:

Helmert 2D – MDT Surveying Manual (aplitop.com)

Log in to reply.