Helmerts mathematics

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.

@olemanriver 

Just 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,

              N₀ and E₀ 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 k₀ = 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)

Transformation methods | Lantmäteriet (lantmateriet.se)

Helmert transform — PROJ 9.2.0 documentation