Lambert Conformal Conic One Standard Parallel Reprojection

[wlkaufman]

Greetings,

 I am trying to reproject .las files from Iowa State Plane to recently developed Iowa Regional Coordinate systems.

 Some of the regions are Transverse Mercator (TM), others are Lambert Conformal Conic one standard parallel (LCC (1SP)). 

 I have been able to reproject to the TM zones successfully using las2las. 

 However for the LCC (1SP) zones it does not seem that las2las supports entry of a scale factor for this type.  It appears that only LCC two standard parallel zones are supported in las2las.  Is this correct, or am I missing a setting? 

The command for LCC is below, it appears to require two parallels. 

-target_lcc 15500000.0 8900000.0 survey_feet 42.65 -92.25 42.65 42.65 -target_elevation_survey_feet

 Parameters:

  Std parallel and grid origin  42.65

  central meridian             -92.25

  false northing                8900000.0

  false easting                15500000.0

  standard parallel scale  1.000032

 Thank you.

[Martin Isenburg]

Hello Geodetic Specialists,

can someone enlighten us how the LCC-1SP with one standard parallel relates to the LCC-2SP with two standard parallels? It is correct that I have only implemented LCC-2SP but I am not sure whether LCC-1SP is simply a subset of  LCC-2SP with one of the two standard parallels at a fixed number of a fixed ratio to the other.

Regards,

Martin @rapidlasso

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[Evon Silvia]

This is a shot in the dark based on 5 minutes poking around in my land surveying text book, but perhaps LCC-1SP simply means that the cone is tangent to the surface of the earth instead of transverse? This would imply that the two standard parallels are equivalent. At least, that's what it means when you have a Mercator projection instead of a Transverse Mercator projection.

I'd have to dive into the math more deeply to discern the computational implications of this, but hopefully someone better informed can contribute. My assumption is that this would simplify the mathematics significantly.

Evon

[Michael Olsen]

Martin,

The best resource i've found for map projections is:"Map Projections: A Working Manual" 1987, Snyder, John P., USGS Professional Paper: 1395.  You can download a pdf here if you don't have it already:  http://pubs.er.usgs.gov/publication/pp1395

Here is a quote from p. 105:

"Two parallels may be made standard or true to scale, as well as conformal.  It is also possible to have just one standard parallel.  Since there is no angular distortion at any parallel (except at the poles), it is possible to change the standard parallels to just one, or to another pair, just by changing the scale applied to the existing map and calculating a pair of standard parallels fitting the new scale."

I need to brush up on my map projections first, but I interpret that to mean that you can adjust the scale and have a pair of standard parallels that are equivalent to the one standard parallel. 

Note that if you run through the math presented in Stewart's manual, equation 15-3 (sphere) or 15-8 (ellipsoid) is calculated as n = sin (phi1) when you have only one parallel (phi1=phi2) since the original is indeteriminate.  The rest only require the one parallel. 

Also, NOAA Manual NOS NGS5 is a good resource, but my guess is you already have that:
http://www.ngs.noaa.gov/PUBS_LIB/ManualNOSNGS5.pdf

I'll forward this email to a colleague, Michael Dennis, who has developed a series of low distortion projection coordinate systems and can shed some light.

(i'll even try to get him on this mailing list now that he took our 3D scanning\LIDAR course here at Oregon State last quarter).

Cheers,

Mike

Michael Olsen
Eric HI and Janice Hoffman Faculty Scholar
Assistant Professor of Geomatics
School of Civil and Construction Engineering
Oregon State University
Email: michae...@oregonstate.edu
Phone:  (541)-737-9327

[Martin Isenburg]

Hello,

Michael Olson brought Michael Dennis into an off-line continuation of this thread. Michael is a current PhD student at Oregon State, on leave from NOAA NGS, and has his own company Geodetic Analysis, LLC.  He has developed several low distortion projection coordinate systems as part of that.  Chris Parrish is his PhD advisor. Below are Michael Dennis' insights and they are good news. An LLC 2SP and be mapped to a corresponding LLC 1SP and vice versa.

Martin

--------------

Hello all,

 

Nice to meet you Martin!  I saw that the Iowa Regional Coordinate System (IaRCS) is mentioned in the first email of this thread.  As it happens, I designed the IaRCS (user manual available at www.iowadot.gov/rtn/pdfs/IaRCS_Handbook.pdf), so hopefully I can help you with the Lambert Conformal Comic (LCC) projection.

 

Let me start by saying that the 1- and 2-parallel LCC are mathematically identical.  Both are defined by the same parameters, one of which is the scale on the standard (central) parallel.  The only difference is that for the 2-parallel, the scale on the central parallel is implicitly defined by the separation between the standard parallels.  The further apart the standard parallels, the smaller the scale.  In contrast, for the 1-parallel LCC, the central parallel and its scale are defined explicitly.  The 2-parallel can be also be scaled in exactly the same manner (although doing so would be odd, since the scale is already defined by the unequal standard parallels).

 

I recommend code be written such that there is only one LCC, which can be specified with either equal or unequal standard parallels.  That is what I do, and it’s done in other software  implementations (such as by Esri).

 

There are many sources for LCC algorithms, two of which were cited by Mike in a previous email in this thread, Stem (1990) and Snyder (1987).  For my implementations, I modified the equation in Stem (including change of west longitude to negative).  On the lower half of p. 27 of Stem (equations are not numbered), there is an equation sin φ_o = ln […]/(Q_n-Q_s).  If the standard parallels are equal, then Q_n = Q_s  and you get division by zero (and as is well known only Chuck Norris can divide by zero).  For the case where standard parallels are equal, instead use sin φ_o = sin φ, where φ is the central latitude.  The same method is stated for eq (15-8) on p. 108 of Snyder for the ellipsoid, which is also the case for the sphere (in Mike’s reply, he already identified this as the solution to the problem).

 

As already noted by Evon, if the standard parallels are equal, then the scale on the standard parallel is exactly 1 (can be visualized as a cone tangent to the ellipsoid or sphere).  It is almost always necessary to also apply a scale factor, since usually the cone is non-intersecting (i.e., above the ellipsoid) when the coordinate system is intended to be at ground (as it is for the IaRCS, ORCS, and other such low distortion systems).  In that case the computed LCC coordinates are multiplied by (scale factor) minus 1.  Exactly the same thing can be done with a 2-parallel Lambert, although it seems kind of silly to do it this way).

 

Mathematically, there really is no need to ever use a 2-parallel LCC; that type of the LCC merely exists by arbitrary convention. Every 2-parallel LCC can be converted to an equivalent 1-parallel version.  I’ve attached a handout from one of my workshops that give some equations for that purpose (on p. 17).  The reverse should also be true, that a 1-parallel can be converted to a 2-parallel [note: email rapidlasso for this handout].  But I have never done this, because it seems unnecessary and counterproductive for 1-parallel LCCs, especially with standard parallel scale > 1.  It may be possible to derive a closed form relationship to do this, but I suspect an iterative solution would be necessary, and I believe there is not a unique solution.  And in any case it would require an additional scale factor, which some implementations of the 2-parallel LCC do not allow.  For example, for the IaRCS Waterloo zone in the initial email, it is impossible to create a 2-parallel version without an additional scale factor to get standard parallel scale = 1.000032.

 

Hope all this helps.  Let me know if you have any comments or questions.

 

Cheers,

 

Michael