Drawing Equirectangular VR Panoramas with Ruler, Compass, and Protractor

Abstract

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 handdrawn 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.

Eq A Sketch 360, a Serious Toy for Drawing Equirectangular Spherical Perspectives

“…In order to understand the purpose and uses of Eq A Sketch 360, one must understand at least the basics of spherical perspective and its relation to VR panoramas. We start this section with a brief review of what can be more leisurely read in [4] and [2]. Then we finish the section with a simple result that seems to be unpublished elsewhere: a calculation of an equirectangular geodesic from two given arbitrary points.…”

“…In [4], the method is both presented and applied to the case of the azimuthal equidistant (fisheye) perspective of Barre and Flocon, generalizing it to a 360-degrees view. In [2], the method is applied to equirectangular spherical perspective. This is an interesting application because equirectangular projection is commonly used by VR software to store 360-degree photographic data to be viewed as VR panoramas.…”

“…Non-vertical geodesics are determined by the position of their apex, i.e., of the highest point reached by the geodesic. Their perspective images have the form [2]…”

“…In [2] it is shown that the geodesics obtained from equation 1 are well approximated by simpler curves such as arcs of circles and sinusoids up to about 60 degrees elevation, and higher elevation ones are obtainable by descriptive geometry constructions. But since such constructions are hard to use in outdoor drawing, a practical gridding method was proposed [2,3], based on a minimal set of geodesics that generates all others by translation. For example, in Figure 2 (top) we see that points A and B are not connected by any of the geodesics of the grid presented.…”

“…Analogously, the present program allows the user to draw any equirectangular line projection using combinations of either horizontal or vertical equirectangular lines. This is possible because these "horizontals" and "verticals" form, by translational symmetry, a generating set for the geodesics of the sphere, as shown in [2]. The program's main feature is the exploitation of this symmetry, using a sliding equirectangular grid and snap-to commands that allows the user to draw with precision an equirectangular line selected by choosing two arbitrary points or a special apex point.…”