Let's Sketch in 360º: spherical perspectives for virtual reality panoramas

Conferência realizada em Stockholm, Sweden de 25–29 julho de 2018In this workshop we will learn how to draw a 360-degree view of our environment using spherical perspective, and how to visualize these drawings as immersive panoramas by uploading them to virtual reality platforms that provide an interactive visualization of a 3D reconstruction of the original scene. We shall show how to construct these drawing in a simple way, using ruler and compass constructions, facilitated by adequate gridding that takes advantage of the symmetry groups of these spherical perspectives. We will consider two spherical perspectives: equirectangular and azimuthal equidistant, with a focus on the former due to its seamless integration with visualization software readily available on social networks. We will stress the relationship between these panoramas and the notion of spherical anamorphosis.info:eu-repo/semantics/publishedVersio

A Construction of the Total Spherical Perspective in Ruler, Compass and Nail

We obtain a construction of the total spherical perspective with ruler, compass, and nail. This is a generalization of the spherical perspective of Barre and Flocon to a 360 degree field of view. Since the 1960s, several generalizations of this perspective have been proposed, but they were either works of a computational nature, inadequate for drawing with simple instruments, or lacked a general method for solving all vanishing points. We establish a general setup for anamorphosis and central perspective, define the total spherical perspective within this framework, study its topology, and show how to solve it with simple instruments. We consider its uses both in freehand drawing and in computer visualization, and its relation with the problem of reflection on a sphere.Comment: Major revision of the 2015 version, with many changes, including and a new title. Main results unaltered, but important changes to the definitions, to notation and organization, and correction of minor errors. Illustrations revised/added, including a major illustration of spherical perspective on page 22. Added references to several works previously unknown. 25 pages, 12 figure

A geometria (descritiva) da anamorfose e das perspectivas curvilíneas

Workshop "Matemática e Arte", realizado na Universidade de Évora de 17-18 de julho de 2017A anamorfose, enquanto artifício mimético, é o conceito fundamental que sustenta a perspectiva clássica e as perspectivas curvilíneas centrais. Enquanto conceito transversal merece uma plataforma, que presentemente não ocupa, no ensino das perspectivas, da geometria descritiva, e da realidade mista em computação gráfica e arte digital. Nesta palestra damos um panorama de vários trabalhos recentes do presente autor que pretendem contribuir para a criação de tal plataforma.Anamorphosis, as an instrument of mimesis, is the foundation both for linear perspective and for all central curvilinear perspectives. It is a transversal concept that merits a central role in the teach ing of perspective(s), of descriptive geometry, and of mixed reality in computer graphics and digital art. In this talk we survey some recent efforts by the present author to establish such a role for anamorphosis.info:eu-repo/semantics/publishedVersio

Anamorphosis: optical games with perspective's playful parent

We explore conical anamorphosis in several variations and discuss its various constructions, both physical and diagrammatic. While exploring its playful aspect as a form of optical illusion, we argue against the prevalent perception of anamorphosis as a mere amusing derivative of perspective and defend the exact opposite view - that perspective is the derived concept, consisting of plane anamorphosis under arbitrary limitations and ad-hoc alterations. We show how to define vanishing points in the context of anamorphosis in a way that is valid for all anamorphs of the same set. We make brief observations regarding curvilinear perspectives, binocular anamorphoses, and color anamorphoses.info:eu-repo/semantics/publishedVersio

Explorations in rational drawing

We discuss the position of the author’s spherical perspective work within a tradition of Rational Drawing, a discipline at the interface of mathematics and the arts.info:eu-repo/semantics/publishedVersio

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

Eq a Sketch 360 is a simple program for raster sketching VR panoramas in equirectangular spherical perspective. It is built as a serious toy, to develop sketching intuition regarding equirectangular drawing as proper perspective drawing, with its specific constructions of vanishing points, geodesics, line projections, antipodes, and grids. It is useful as a teaching aid and as a production tool for preliminary perspective sketches to be further rendered with other digital or traditional tools. It is naturally adapted for the input variables adequate for observational sketches. In this paper we survey the operation and purposes of the program. We also show how it calculates the equirectangular geodesic through two given points, which enables one of its main drawing features.I was sitting for quite a while on the result of section 2.3 and its application. My thanks to Michael Scherotter of Microsoft, the author of Journalist and Sketch 360, and a fellow urban sketcher, whose questions pushed me to get on with it and write it down. I hope Michael and other developers will find the result useful for their own software and art. I was financially supported by Portuguese national funds through FCT project UID/Multi/04019/2013.info:eu-repo/semantics/publishedVersio

Anamorphosis reformed: from optical illusions to immersive perspectives

We discuss a definition of conical anamorphosis that sets it at the foundation of both classical and curvilinear perspectives. In this view, anamorphosis is an equivalence relation between three-dimensional objects, which includes two-dimensional representatives, not necessarily flat. Vanishing points are defined in a canonical way that is maximally symmetric, with exactly two vanishing points for every line. The definition of the vanishing set works at the level of anamorphosis, before perspective is defined, with no need for a projection surface. Finally, perspective is defined as a flat representation of the visual data in the anamorphosis. This schema applies to both linear and curvilinear perspectives and is naturally adapted to immersive perspectives, such as the spherical perspectives. Mathematically, the view here presented is that the sphere and not the projective plane is the natural manifold of visual data up to anamorphic equivalence. We consider how this notion of anamorphosis may help to dispel some long-standing philosophical misconceptions regarding the nature of perspective.info:eu-repo/semantics/publishedVersio

Workshop - Drawing equirectangular perspectives for VR panoramas with Eq A Sketch 360

In this workshop we will explore the construction of immersive environments in equirectangular perspective using the Eq A Sketch 360 software. Eq A Sketch 360 is a serious toy for spherical perspective drawing. It has two innovative features: a sliding geodesic grid and an equirectangular snap-to ruler. These tools turn equirectangular drawing into a proper perspective, where all lines and vanishing points may be drawn by hand, to create immersive environments from either observation or imagination. This contrasts with previous methods of equirectangular drawing, that either avoided perspective altogether by drawing directly in VR view, or were limited to fixed grid methods with ad-hoc estimation of measurements. Eq A Sketch both forces and helps the user to learn spherical perspective. We will show how to draw by hand with perfect control of proportions and bearings, to make standalone designs or constructions that can be mixed with 360-degree photography.Author funded by FCT national funds project UIDB/Multi/04019/2020info:eu-repo/semantics/publishedVersio

A GeoGebra tool for drawing immersive perspectives

Spherical perspective provides a connection between traditional handmade drawings and virtual reality environments, which can be exploited to create new forms of hybrid immersive artworks. We discuss the use of GeoGebra tools as auxiliary software for the creation of immersive designs in azimuthal equidistant (360-degree fisheye) spherical perspective.Author A. B. Araújo was funded by FCT national funds through project UIDB/Multi/04019/2020.info:eu-repo/semantics/publishedVersio

Guidelines for drawing immersive panoramas in equirectangular perspective

Virtual Reality (VR) Panoramas work by interactively creating immersive anamorphoses from spherical perspectives. These panoramas are usually photographic but a growing number of artists are making hand-drawn equirectangular perspectives in order to visualize them as VR panoramas. This is a practice with both artistic and didactic interest. However, these drawings are usually done by trial-and-error, with ad-hoc measurements and interpolation of precomputed grids, a process with considerable limitations.We develop in this work the analytic tools for plotting great circles, straight line images and their vanishing points, and then provide guidelines for achieving these constructions in good approximation without computer calculations, through descriptive geometry diagrams that can be executed using only ruler, compass, and protractor.info:eu-repo/semantics/publishedVersio