ArcMap - Calculation Based Projections

[ChuckTurlington]

Hello,

I am using ArcMap 10.4.1.

I am still studying the concepts of projections.  I understand the ideas behind the developable surfaces and learned from several books that the surfaces are just an idealized way to explain the concepts of projections but that the developable surfaces are not literal in the calculation of a projection.  It seems to me that the math behind a projection would be like a vector intersecting the developable plane, normal to the surface, and the calculated change in location from an initial lat-long position, gives the new projected location.  However, if the developable surface is not literal in the projection calculation, it does not make sense without some consideration for the developable surface, it seems to me.  Even to use 3 dimensional trigonometry, treating a position on elliptical datum and the new point position on the developable surface, considers the developable surface in the projection.  Is it possible that the projection calculation really does utilize the developable surface, and the claim that it does not, is only in reference to Grid Based projections ?

[DanPatterson_Retired]

Chuck... there are dozens of pages on the mathematics behind projections not to forget the dozens of books.  Is there a particular mathematical presentation that you are grappling with?

[ChuckTurlington]

Hi Dan,

Thanks again for your feedback and expertise on this subject.  I am aware of some of the formulae describing the calculation based projections, very long trig expressions, but no particular method in mind.  But as a generalization, I am interested to know if my novice understanding makes sense without having to become a geodesy major.  I am completing a 2-3 year research project for my own benefit as a civil design engineer, but it does not seem complete without confirmation of the above ideas, that the developable surface is a large part of the calculation...

[MelitaKennedy · Accepted solution]

Once you start using ellipsoids rather than spheres, it's much more difficult to directly use a developable surface. It is a good approximation, particularly for understanding in general how projections work. 

Think about projections like the Eckerts. I call them cylindrical but they're actually pseudocylindrical. Eckert IV and VI have poles-as-lines that are a percentage of the equator so right there they're not truly cylindrical. Adjustments like that are the rule rather than exception for most projections because otherwise you have very simple projections that really don't do much.

Melita