Drawing Equirectangular VR Panoramas with Ruler, Compass, and Protractor

Keywords: Visual arts, immersive art, immersive panoramas, VR panoramas, equirectangular perspective, spherical perspective, perspective, drawing, descriptive geometry, digital art, ruler and compass, social networks


This work presents a method for drawing Virtual Reality panoramas by ruler and compass operations. VR panoramas are immersive anamorphoses rendered from equirectangular spherical perspective data. This data is usually photographic, but some artists are creating hand-drawn equirectangular perspectives to be visualized in VR. This practice, that lies interestingly at the interface between analog and digital drawing, is hindered by a lack of method, as these drawings are usually done by trial-and-error, with ad-hoc measurements and interpolation of pre-computed grids, a process with considerable artistic limitations. I develop here the analytic tools for plotting all great circles, line images and their vanishing points, and then show how to achieve these constructions through descriptive geometry diagrams that can be executed using only ruler, compass, and protractor. Approximations of line images by circular arcs and sinusoids are shown to have acceptable errors for low values of angular elevation. The symmetries of the perspective are studied and their uses for improving gridding methods are discussed.

Author Biography

António Bandeira Araújo, Department of Science and Technology, Universidade Aberta, Portugal
António Bandeira Araújo is a geometer and draughtsman. He holds a degree in physics and a Ph.D. in mathematics. He had training in drawing, painting, and scientific illustration, and has published illustration work. His main interest lies in the interactions between mathematics and the visual arts, both classical and digital. He lectures at Universidade Aberta in Portugal and is a member of CIAC-UAb (Centro de Investigação em Artes e Comunicação) and CMAF-CIO (Centro de Matemática, Aplicações Fundamentais e Investigação Operacional).


Anderson, D. (n.d.). panorama illustrations. Retrieved April 19, 2017, from https://www.flickr.com/photos/davidanderson/sets/72157627385462893

Araújo, A. (2015). A Construction of the Total Spherical Perspective in Ruler, Compass and Nail. Retrieved from http://arxiv.org/abs/1511.02969

Araújo, A. (2017). Guidelines for drawing immersive panoramas in equirectangular perspective. In C. Caires (Ed.), Artech 2017. International Conference on Digital Arts, 8 - "Interfaces of Tomorrow". Macau: ACM.


Araújo, A. (2017a). Anamorphosis: Optical Games with Perspective's Playful Parent. In J. N. Silva (Ed.), Recreational Mathematics Colloquium V G4G (pp. 71-86). Lisbon: Ludus.

Araújo, A. (2017b). Cardboarding Mixed Reality with Dürer Machines. 5th Conference on Computation, Communication, Aesthetics & X (pp. 102-113). Lisbon: Univ. do Porto. Retrieved from http://2017.xcoax.org/xCoAx2017.pdf

Araújo, A. (2017c). Notes and materials on equirectangular perspective and VR panoramas. Retrieved from http://www.univ-ab.pt/~aaraujo/equirectangular.html

Art Panorama Group. (2017, April 19). Art Panoramas. Retrieved from Flickr: https://www.flickr.com/groups/artpanoramas/pool/

Barre, A., & Flocon, A. (1964). La perspective Curviligne. Paris: Flammarion.

Barre, A., Flocon, A., & Bouligand, G. (1964). Étude comparée de différentes méthodes de perspective, une perspective curviligne. Bulletin de la Classe des Sciences de La Académie Royale de Belgique, Série 5, Tome L.

Huhtamo, E. (2013). Illusions in motion - media archaeology of the moving panorama and related spectacles (1st ed.). London: The MIT Press.

Kemp, M. (1990). The Science of Art. New Haven and London: Yale University Press.

Michel, G. (2007). L'oeil, au Centre de la Sphere Visuelle. Boletim da Aproged, 3-14.

Michel, G. (2017, April 19). Dessin à main levéee du Cinéma Sauveniére. Retrieved from http://autrepointdevue.com/blog/wp-content/vv/vv-gm-sauveniere/vv-gm-sauveniere.html

Papert, S., & Turkle, S. (1991). Epistemological Pluralism. In I. a. Harel (Ed.), Constructionism (pp. 161-191). Ablex Publishing Co.

Pozzo, A. (1700). Perspectiva pictorum et architectorum, vol 2. Rome.

Rodriguez, H. (2016). Theorem 8.1. 4th Conference on Computation, Communication, Aesthetics and X, (pp. 333-340). Bergamo.

Snyder, J. (1987). Map Projections - a Working Manual (Vol. 1935). Washington, U.S.A.: U.S.G.P.O.

Stiegler, B. (2010). For a New Critique of Political Economy. Cambridge: Polity Press.

Termes, D. (n.a.). New Perspective Systems. self-published.

Equirectangular perspective of a stairwell