How Many UTM Cylinders are there?

Question

I've been reviewing the theory behind the UTM projection and I think I've gotten a handle on the approach, including the difference between transverse secant cylinders and tangent cylinders, standard lines and central meridians.

For any given single UTM projection, the zone only seems to apply to one side of the secant cylinder. Does each UTM zone have its own secant cylinder, or does one cylinder include opposite sides of the reference sphere?

I've been through many of the available sources on the net, and looked through a couple GIS text books, without a clear answer. I found the Penn State GIS textbook helpful https://www.e-education.psu.edu/natureofgeoinfo/c2_p23.html as well as the USGS Map Projections Working Handbook https://pubs.usgs.gov/pp/1395/report.pdf but I've not yet found a source that claims a solid answer.

I'd be happy with a "Let me Google that for you" answer... I've had no luck.

Answer 1

In general, the transverse Mercator algorithms on an ellipsoid only support one hemisphere. The TM algorithms on a sphere could show both "sides" of the cylinder but usually don't as people don't like to see upside down countries except in polar/azimuthal projections.

Answer 2

I found two references that I think lock in the most likely answer at 60 cylinders, but a close reading of the definition, could reasonably lead to an interpretation of 30, 60, or 120.

If one ignores the potential for flipping the opposite side map (antipodal?) then 30 cylinders could be the smallest number, as mkennedy spelled out above.

The 60 cylinders model is the most common approach used to visualize them, but interestingly enough, the two references only mention 120 Spatial Reference Frames or instances.

First up, the US National Geospatialintelligence Agency (NGA) defines UTM as follows:

UTM is a family of 120 instances of the general form of the transverse 
Mercator projection. Each instance is called a zone and is given a 
zone number Z between -60 and +60 excluding zero. 

Source:NATIONAL GEOSPATIAL-INTELLIGENCE AGENCY STANDARDIZATION DOCUMENT Implementation Practice The Universal Grids and the Transverse Mercator and Polar Stereographic Map Projections 2014-03-25, NGA.SIG.0012_2.0.0_UTMUPS, Section 7.1 page 32.

Second, the International Standards Organization (ISO) group has published the standard for Spatial Reference Models, and it defines UTM as:

A set of 120 localized Spatial Reference Frames (SRFs), 
where limited overlap is modelled by extended validity regions in the member SRFs. 

Source: International Standard ISO / IEC 18026, Information Technology: Spatial Reference Model, ISO / IEC 18026, Section 8.7.7 - Table 8.6.0, page 192.

So, one could also make a case for 120 cylinders (half cylinders?) since each UTM zone could be aligned to a unique one. If the mappers use different ellipsoids for their UTM projections, then the cylinders will naturally be different too.

I think the UTM standard for the US to only use the WGS 1984 ellipsoid - and that is the standard. The referenced NGA document also spelled out approaches for TM to be applied to many other ellipsoids, starting at page 20.

I hope this helps - corrections welcome.