Home Forums General Discussion Drawing in Spherical Perspective (free access until august 31st at the JMA)

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    António Araújo
    António Araújo
    Forum Member


    my name is António Araújo, I do research on spherical perspectives and VR panoramas at Aberta University, in Portugal. I just registered and was wondering if anyone here is interested in drawing spherical perspectives, either “fish-eye” (azimuthal equidistant), equirectangular, or others, by hand or with elementary tools such as ruler and compass. If so, do say hello and let me know what you work on. :)

    In case you are working in the subject or just curious: I just published a method to do 360-degree fisheye (azimuthal equidistant) perspective by ruler and compass, in the Journal of Mathematics and the Arts. The publisher has made the paper free access until the end of August, so for now you can download the pdf for free, here:

    Ruler, Compass and Nail: Constructing a Total Spherical Perspective

    Also I published another paper on how to do equirectangular perspectives by hand, with a method which is not as pretty,  but is rather practical to transform into a VR panorama since, as you all know, equirectangular is the standard format for VR visualization. That paper you can download at your leisure since it is permanently free access:

    Drawing Equirectangular VR Panoramas with Ruler, Compass, and Protractor

    The point in both papers is to be able to find and draw all lines and vanishing points almost as easily and precisely as in classical perspective, therefore expanding the drawing methods of both artists and architects. The point is not to do away with the computer but to interact with it in interesting ways that enhance traditional drawing instead of replacing it.

    Anyway, enough with the shameless plug :); if you are doing any work on the field or are just interested in chatting about it then let me know.

    All the best,



    Tony Meneses
    Forum Member

    I had seen some illustrations on 360 panoramic sites but never have I seen anyone publish the fundamentals. Thank you for your scholarly publication, I look forward to reviewing it more thoroughly.

    António Araújo
    António Araújo
    Forum Member

    Dear Tony,

    I look forward to your comments!

    Regarding the first paper, in case you didn’t get a copy while it was free access, and if you don’t have access to that journal, you can just email me and I’ll send you a free copy (my email is antonio.araujo (at) uab.pt – btw, the offer is valid to anyone who may be interested)

    As far as I know, there were no papers dealing with the fundamentals, except for the case of 180º fisheye (solved in Barre and Flocon’s 1960’s classical book “La perspective Curviligne”). A bit later there were a couple of papers in Leonardo that tried to solve the 360º case and failed (but somehow were interpreted as succeeding, basically because not many people were able to interpret the problem at the time). Meanwhile, a brilliant Belgian architect called Gerard Michel actually knew how to draw the 360º degree case but didn’t publish it until very recently. His method was grid-based, though, so not a complete solution in the sense of Flocon and Barre’s solution, but much, much better than the stuff that got published in Leonardo (and moreover he did beautiful drawings with his method). Apparently Gerard knew nothing about either the Leonardo’s papers or even Flocon and Barre. When I did my solution I on the other hand knew Flocon and Barre’s work but was ignorant of both the works in Leonardo and the work by Michel. :) Fortunately I met Gerard on facebook (of all places) in time to give him due credit in the published paper (and also found out about the other papers on Leonardo). Basically the history of fisheye perspective (technically, azimuthal equidistant spherical perspective) sure is a mess of partial solutions, misconceptions, and people reinventing the wheel. :) I guess that comes from being a rather rarified field, with just a few enthusiasts actually working on it seriously, and most of them being artists who didn’t have the right mathematical tools.

    As for the equirectangular case, it is newer, and I guess nobody did it because most people interested in it come from computer science and so they don’t think of it as an actual perspective, where vanishing points are the essential objects. From a computer rendering point of view you can just brute force it, and vanishing points aren’t explicitly considered – parallel lines will still converge because, well, you are rendering the hell out of the scene, pixel-by-pixel. :) But if you want to draw by hand then you have to think of it in a different way, reducing lines to understandable projections that can be done in at most a few (rather than a few thousand) operations.

    Anyway, looking forward to your views on this.

    All the best,

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