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Cartesian Coordinates – We Can Do Better

grpahicof Cartesian Coordinates - We can Do Better
Cartesian Coordinates - We can Do Better

Rene Descartes passed away midway through the 17th century but his system for ordering the natural world into 3 planes continues to serve as the fundamental building block of the geospatial industry to this day. The Cartesian Coordinates of X,Y, Z are of basic and critical importance to both pioneering geospatial professionals and entry level algebra students.

There are 3 ideas to unpack here. The first being that Descartes was a brilliant scientist who devised a logical and beautifully simple system for ordering the unordered natural world around him. By inventing Cartesian coordinates Descartes was able to nominally measure in 3-dimensions and thusly organize phenomena into spatial planes; the X, Y and Z. Descartes’ system is so effective that it is in use in nearly every facet of the geospatial industry today in a format that Descartes himself would recognize as unchanged since his day.

The 2nd idea for your consideration is that Cartesian coordinates are, as is tradition in cartography, representations of a lie of sorts. In a science that has traditionally warped reality to fit 3-dimensions into 2-dimensional reports these lies are nothing new or terribly out of the ordinary, but the effort to which geospatial scientists go to place locational band-aids on Cartesian coordinates to make them closer representations of reality are extraordinary.

The inherent flaw that Descartes did not solve for is that Cartesian space is cubed. So while we can use Cartesian space to create fantastically simplified representations of the real world that preserve most of the information about spatial phenomena, we have to then by hook or crook jam this information into little boxes. Literally, little boxes.

This is all well and good to the bulk of programmers working with or in the geospatial industry; I can code all location information into just three data fields! Need to compute directions or distance from Point A to B? Just need to operate on three data fields to get an accurate return! The ugly truth hiding behind this simplicity is the world of datums, projections and spatial reference systems, all of which serve to make sense of your unrealistic 3-dimensional coordinates and place them closer to reality.

The third facet of our trip through Cartesian space is answering the question if Cartesian coordinates are a lie, how do we tell the truth? I believe the solution to this error lies in an idea first discussed in the 1990’s by a little known Denver-area professor and subsequently discarded as being too complex: a regular dodecahedral spatial reference system.

Dr. Joseph K. Berry first raised the idea of coding a regular dodecahedral system to improve upon the voxel (cubed) system of Descartes, but ran into the issue you’d expect going from a 3-string location system to a minimally 13-string location system; complexity is a deterrent to adopters. At the time that Dr. Berry was first raising this topic computational bandwidth was also a deterrent and consideration, but now with even the simplest and cheapest Arduino processors providing more kick than the supercomputers of yesteryear the challenge of computing an additional 10+ fields for every geospatial operation fades as a roadblock.

So why would or should geographers tackle the challenge of developing a regular dodecahedral spatial reference system? To my mind it’s a worthwhile pursuit for a variety of scientific and practical reasons. The science of location and locational phenomena is ever-changing and using a framework from nearly 400 years ago seems a bit out of, ahem, place. In any industry there exists a naturally occurring stasis to stick to what is known and geography is no different.

Would our industry even be recognizable without the base assumption that every measurement can be dumped into one or more of X, Y and Z? Put it this way: I operate a terrestrial LiDAR scanner, scanning millions of real world vertices with previously unknown real-world accuracy, and when I get back to the office I assemble that data into a framework that badly mangles the real-world location of these phenomena in an effort to make my job manageable. With the greatest respect to Rene Descartes, we can do better.

Thank you to Andrew Mallin for this thoughtful discussion.

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  • It’s not clear to me how dodecahedrons tessellate or eliminate the problems discussed, and I can’t find any material online aside from this article. Is there a paper or website documenting this idea in further detail?

  • Why not just go full out spherical? radial distance, polar angle, and azimuthal angle using center of earth as 0,0,0 so as datasets overlap & merge into global smart communities it all jives. Can we all go on universal time too? One giant simultaneous NYE party! I applaud the discussion and look forward to following it…

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