Is the quadrilateralized spherical cube map projection [1] the same as Snyder's cubic equal area map projection [2]?

I know both are equal area polyhedral map projections ( http://www.progonos.com/furuti/MapProj/Dither/ProjPoly/projPoly.html ) based on the cube, but i can't access either publication to answer the question myself.


[1] E. M. O’Neill and R. E. Laubscher, Extended studies of the quadrilateralized spherical cube earth data base, Tech. Report 3-76, Computer Sciences Corp Silver Spring Md System Sciences Div, 1976.

[2] Snyder, J.P. 1992. An equal-area map projection for polyhedral globes. Cartographica 29(1): 10-21.



(1) Yes I am the F. Kenneth Chan in the 1975 Navy report (114 pages) which is now out-of-print. I have one of the original 10 copies printed. Fort Belvoir in Virginia (USA) has a copy. So does the Australian Defence Department In Canberra. However, the National Technical Information Service (NTIS) in Springfield, VA can reproduce copies. Also, I have come across a "Print-on-Demand" purchase capability on Amazon. I also wrote an abridged paper (25 pages) which I can send you if you give me your address.

(2) No, the projections of Reference [1] is not the same as that in Reference [2]. You have seen the Laubscher projection [1] above. The Snyder projection [2] may be seen in the following link: Architecture+_Urban_Planning_files/Cubo%20Final%20Webpage.pdf">http://www.susanaarellano.com/susanaarellano/Susana_Arellano_Alvarado_Architecture+_Urban_Planning_files/Cubo%20Final%20Webpage.pdf

(3) Unbeknownest to me after I left Computer Sciences Corporation in 1975, Laubscher desecrated my work. It was only about two years ago that I found out what he did. He is dead now. It is unfortunate that the damage he did still lives on.

(4) More applications of the QLSC may be found by going to pages 7 and 8 of the conference program in the following link: http://www.space-flight.org/docs/2012_winter/final_program.pdf


  • Alex, The link above to the Snyder projection does not seem to work. Go to Google and look up "John Snyder Equal Area Polyhedra". The item by Susana Arellano is on the second or third page. Ken
    – Ken Chan
    Nov 14 '12 at 1:50
  • Thanks for your help, Ken. I'm in the process of getting NTIS PDFs of your 1975 report and Laubscher's 1976 report. I was also pointed to [4] for summaries of your and Laubscher's projections, where [4] = Calabretta, M. and Greisen, E.: Representations of celestial coordinates in FITS. Astronomy & Astrophysics 395, 3, 1077–1122, 2002.
    – araichev
    Nov 14 '12 at 21:31
  • 1
    Alex, When you get my report, you will notice that my direct and inverse mappings are accurate to 4 or 5 significant figures. Laubscher changed my coefficients in his report. A comparison reveals that the transformations given in Calabretta's paper do not bear any resemblance to mine. The coefficients are not from my original report but are based on Laubscher's report. Because of this, Calabretta states that my transformations are accurate to 1%. Moreover, it is also stated that the code disagrees with the formulas. Please bear this in mind. My advice is to go by the report in the original for
    – user12795
    Nov 15 '12 at 4:38
  • Ken, please consult our faq about how this site works. We're not a chat or discussion site, so multiple replies that are basically comments are inappropriate here. A good solution for handling follow-ups is to edit your original post, unless you are creating a genuinely new and different answer to the question.
    – whuber
    Nov 15 '12 at 6:13

(1) Your Reference [1] is misleading. O'Neill (programmer) and Laubscher (analyst) were not the first ones to come up with the Quadrilateralized Spherical Cube (QLSC). The original work was done by Chan (analyst) and O'Neill (programmer) in 1973 and published as a Navy report in 1975. Go to Google and Wikipedia for "Quadrilateralized Spherical Cube".

(2) The work of O'Neill and Laubscher did not truly "quadrilateralize" a sphere but actually "triangularized" it. As a result, there are mathematical singuarities along the diagonals of the cube and, hence, also at the poles. This is evident from the sharp breaks in the latitudes. Even their longitudes are evidently far from being straight. On the other hand, the earlier work of Chan and O'Neill did not have such singularities and the cube is truly quadrilateralized. This is evident in the smoothness of the latitudes and the straightness of the longitudes. See diagrams in following link: http://www.progonos.com/furuti/MapProj/Dither/ProjPoly/projPoly2.html

(3) There is no exact closed form solution for the QLSC mapping. The original work of Chan and O'Neill expressed the mapping as a truncated convergent infinite series (which is the norm when dealing with infinite series). Because of this, Laubscher erroneously referred to it as "approximate" and claimed his own work as "exact" even though it was not a QLSC. This point was buried in the fuss raised by Laubscher and escaped the notice of the general public.

(4) The original QLSC was adopted for use by the NASA Cosmic Background Explorer (COBE). Results from COBE demonstrated the anisotropy of the cosmic microwave background radiation only moments after the Big Bang. A brief description of the QLSC is given in the link: http://adsabs.harvard.edu/full/1992ASPC...25..379W

(5) Additional history of the QLSC may be found in the "Quadrilateralized Spherical Cube Forum". Look up Google with these keywords.

(6) John Snyder's equal area projection onto a spherical cube is not quadrilateralized and also has mathematical singularities evidently along the diagonals of the cube and the poles. Go to Google and look up "John Snyder Equal Area Polyhedra".

  • 2
    Welcome to our site! Thank you very much for that very well informed reply.
    – whuber
    Nov 13 '12 at 18:55
  • Thanks for your detailed answer, @KenChan. Are you the Chan of [3]:= F.K. Chan and E.M. O'Neill. 1975. Feasibility Study of a Quadrilateralized Spherical Cube Earth Data Base, Computer Sciences Corporation, EPRF Technical Report 2-75 (CSC). Monterey, California: Environmental Prediction Research Facility? Things are getting clearer: the projection of [3] is different from the projections of [1] and [2]. Going back to my original question, are the projections of [1] and [2] the same? Thanks again.
    – araichev
    Nov 13 '12 at 21:23

The original technical report [3] is available for free and in PDF form at: https://ntrl.ntis.gov/NTRL/dashboard/searchResults/titleDetail/ADA010232.xhtml

[3] Chan, F. K.; O'Neill, E. M. "Feasibility Study of a Quadrilateralized Spherical Cube Earth Data Base." Computer Sciences Corp., Silver Spring, Md. System Sciences Div; Environmental Prediction Research Facility (Navy), Monterey, Calif., Final rept. May 1974-Mar 1975.

The 25-pages abridges document mentioned by Chang himself, seems to be the available at:


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