Global projection with zero scale distortion at non-zero latitude

Question

I am looking for a global projection with zero scale distortion at non-zero latitude.

I have tried using an Oblique Mercator projection centred at a specific latitude and longitude of centre (with Mercator orientation) in GeoTools (specified using WKT) but it seems to be a standard Mercator projection in terms of scale distortion. I was hoping it might have zero scale distortion at the latitude of centre but this doesn't seem to be the case.

Does such a projection exist? If so, what is the WKT for it?

Answer 1

An Oblique Mercator will have near-zero distortion along a great circle of contact; that excludes all circles of latitude (except the Equator).

Projections that minimize distortion along circles of latitude tend to be derived from conic projections and their limiting forms, cylindrical projections. The conic projections are usually good only for hemispheres. (You could create an interrupted conic, perhaps joining one for the Northern hemisphere with a similar one for the Southern hemisphere along the Equator, but creating such custom projections would take us beyond the scope of this question, as well as outrunning the capabilities of all GISes.)

You can always modify a cylindrical projection to have true scale along any circle of latitude, simply by stretching or compressing it vertically. This can be done with the Mercator (the usual one).

PROJCS["Sphere_Mercator",GEOGCS["GCS_Sphere",DATUM["D_Sphere",SPHEROID["Sphere",6371000.0,0.0]],PRIMEM["Greenwich",0.0],UNIT["Degree",0.0174532925199433]],PROJECTION["Mercator"],PARAMETER["False_Easting",0.0],PARAMETER["False_Northing",0.0],PARAMETER["Central_Meridian",0.0],PARAMETER["Standard_Parallel_1",72.0],UNIT["Meter",1.0]]

To confirm its accuracy, note that on the sphere one degree of longitude at 72 degrees latitude is the same length as cos(72 degrees) = 1/(1 + Sqrt(5)) = 0.309 degrees of latitude, whence the spacing on this ten-degree graticule ought to give rectangles with aspect ratios approximately 1+sqrt(5) = 3.24 near 72 degree latitude (north and south): that is indeed the appearance here.

Alternatively--to avoid the expansion of the map to infinity at the poles--you could use, say, an Equidistant Cylindrical projection. Here is one with its latitude of scale set to 72 degrees:

PROJCS["Sphere_Equidistant_Cylindrical",GEOGCS["GCS_Sphere",DATUM["D_Sphere",SPHEROID["Sphere",6371000.0,0.0]],PRIMEM["Greenwich",0.0],UNIT["Degree",0.0174532925199433]],PROJECTION["Equidistant_Cylindrical"],PARAMETER["False_Easting",0.0],PARAMETER["False_Northing",0.0],PARAMETER["Central_Meridian",0.0],PARAMETER["Standard_Parallel_1",72.0],UNIT["Meter",1.0]]

In this map, the aspect ratios of the rectangles in the graticule are a constant 1+sqrt(5).