Ruler, compass, and nail: constructing a total spherical perspective

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 Then we finish the section with a simple result that seems to be unpublished elsewhere: a calculation of an equirectangular geodesic from two given arbitrary points.…”

“…Spherical perspectives are maximally symmetric repositories of anamorphic visual data: they are drawings that represent, in a bounded region of a plane, all the visual data accessible to an observer that is stuck to a point in space but free to look around it in any direction [4]. VR panoramas use this data to recreate the immersive visual experience of being inside the original drawn environment.…”

“…Extensions to a full view where attempted through qualitative [9], ad-hoc [10] or grid-based [11] approaches. The full generalization of Barre and Flocon's method to the total sphere was obtained by the present author in [4] where a general schema for spherical perspectives is proposed, based on the separation between two steps with dual roles: anamorphosis and flattening. In the anamorphosis step we project all the visual information onto a geometrical object that is naturally visually immersive and maximally symmetric: we get exactly two vanishing points for each line.…”

“…To do such a rendering by hand requires judicious classifications of vanishing points, geodesics and lines, and proper use of symmetry groups [3], to do in a few steps what a computer does in millions. The purpose of the formalization of spherical perspective in [4] was to give a framework and strategy to solve any such perspective with a view to handmade drawing, through a small number of operations that a human can perform with simple tools such as ruler and compass.…”

“…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.…”