From on-line sketching to 2D and 3D geometry: A fuzzy knowledge based system

The paper describes the development of a fuzzy knowledge based prototype system for conceptual design. This real time system is designed to infer user’s sketching intentions, to segment sketched input and generate corresponding geometric primitives: straight lines, circles, arcs, ellipses, elliptical arcs, and B-spline curves. Topology information (connectivity, unitary constraints and pairwise constraints) is received dynamically from 2D sketched input and primitives. From the 2D topology information, a more accurate 2D geometry can be built up by applying a 2D geometric constraint solver. Subsequently, 3D geometry can be received feature by feature incrementally. Each feature can be recognised by inference knowledge in terms of matching its 2D primitive configurations and connection relationships. The system accepts not only sketched input, working as an automatic design tools, but also accepts user’s interactive input of both 2D primitives and special positional 3D primitives. This makes it easy and friendly to use. The system has been tested with a number of sketched inputs of 2D and 3D geometry

A Fisher-Rao metric for paracatadioptric images of lines

In a central paracatadioptric imaging system a perspective camera takes an image of a scene reflected in a paraboloidal mirror. A 360° field of view is obtained, but the image is severely distorted. In particular, straight lines in the scene project to circles in the image. These distortions make it diffcult to detect projected lines using standard image processing algorithms. The distortions are removed using a Fisher-Rao metric which is defined on the space of projected lines in the paracatadioptric image. The space of projected lines is divided into subsets such that on each subset the Fisher-Rao metric is closely approximated by the Euclidean metric. Each subset is sampled at the vertices of a square grid and values are assigned to the sampled points using an adaptation of the trace transform. The result is a set of digital images to which standard image processing algorithms can be applied. The effectiveness of this approach to line detection is illustrated using two algorithms, both of which are based on the Sobel edge operator. The task of line detection is reduced to the task of finding isolated peaks in a Sobel image. An experimental comparison is made between these two algorithms and third algorithm taken from the literature and based on the Hough transform

Tele-autonomous control involving contacts: The applications of a high precision laser line range sensor

The object localization algorithm based on line-segment matching is presented. The method is very simple and computationally fast. In most cases, closed-form formulas are used to derive the solution. The method is also quite flexible, because only few surfaces (one or two) need to be accessed (sensed) to gather necessary range data. For example, if the line-segments are extracted from boundaries of a planar surface, only parameters of one surface and two of its boundaries need to be extracted, as compared with traditional point-surface matching or line-surface matching algorithms which need to access at least three surfaces in order to locate a planar object. Therefore, this method is especially suitable for applications when an object is surrounded by many other work pieces and most of the object is very difficult, is not impossible, to be measured; or when not all parts of the object can be reached. The theoretical ground on how to use line range sensor to located an object was laid. Much work has to be done in order to be really useful

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

The Three-Dimensional Circumstellar Environment of SN 1987A

We present the detailed construction and analysis of the most complete map to date of the circumstellar environment around SN 1987A, using ground and space-based imaging from the past 16 years. PSF-matched difference-imaging analyses of data from 1988 through 1997 reveal material between 1 and 28 ly from the SN. Careful analyses allows the reconstruction of the probable circumstellar environment, revealing a richly-structured bipolar nebula. An outer, double-lobed ``Peanut,'' which is believed to be the contact discontinuity between red supergiant and main sequence winds, is a prolate shell extending 28 ly along the poles and 11 ly near the equator. Napoleon's Hat, previously believed to be an independent structure, is the waist of this Peanut, which is pinched to a radius of 6 ly. Interior to this is a cylindrical hourglass, 1 ly in radius and 4 ly long, which connects to the Peanut by a thick equatorial disk. The nebulae are inclined 41\degr south and 8\degr east of the line of sight, slightly elliptical in cross section, and marginally offset west of the SN. From the hourglass to the large, bipolar lobes, echo fluxes suggest that the gas density drops from 1--3 cm^{-3} to >0.03 cm^{-3}, while the maximum dust-grain size increases from ~0.2 micron to 2 micron, and the Si:C dust ratio decreases. The nebulae have a total mass of ~1.7 Msun. The geometry of the three rings is studied, suggesting the northern and southern rings are located 1.3 and 1.0 ly from the SN, while the equatorial ring is elliptical (b/a < 0.98), and spatially offset in the same direction as the hourglass.Comment: Accepted for publication in the ApJ Supplements. 38 pages in apjemulate format, with 52 figure

Oval Domes: History, Geometry and Mechanics

An oval dome may be defined as a dome whose plan or profile (or both) has an oval form. The word Aoval@ comes from the latin Aovum@, egg. Then, an oval dome has an egg-shaped geometry. The first buildings with oval plans were built without a predetermined form, just trying to close an space in the most economical form. Eventually, the geometry was defined by using arcs of circle with common tangents in the points of change of curvature. Later the oval acquired a more regular form with two axis of symmetry. Therefore, an “oval” may be defined as an egg-shaped form, doubly symmetric, constructed with arcs of circle; an oval needs a minimum of four centres, but it is possible also to build polycentric ovals. The above definition corresponds with the origin and the use of oval forms in building and may be applied without problem until, say, the XVIIIth century. Since then, the teaching of conics in the elementary courses of geometry made the cultivated people to define the oval as an approximation to the ellipse, an “imperfect ellipse”: an oval was, then, a curve formed with arcs of circles which tries to approximate to the ellipse of the same axes. As we shall see, the ellipse has very rarely been used in building. Finally, in modern geometrical textbooks an oval is defined as a smooth closed convex curve, a more general definition which embraces the two previous, but which is of no particular use in the study of the employment of oval forms in building. The present paper contains the following parts: 1) an outline the origin and application of the oval in historical architecture; 2) a discussion of the spatial geometry of oval domes, i. e., the different methods employed to trace them; 3) a brief exposition of the mechanics of oval arches and domes; and 4) a final discussion of the role of Geometry in oval arch and dome design

Do-It-Yourself Single Camera 3D Pointer Input Device

We present a new algorithm for single camera 3D reconstruction, or 3D input for human-computer interfaces, based on precise tracking of an elongated object, such as a pen, having a pattern of colored bands. To configure the system, the user provides no more than one labelled image of a handmade pointer, measurements of its colored bands, and the camera's pinhole projection matrix. Other systems are of much higher cost and complexity, requiring combinations of multiple cameras, stereocameras, and pointers with sensors and lights. Instead of relying on information from multiple devices, we examine our single view more closely, integrating geometric and appearance constraints to robustly track the pointer in the presence of occlusion and distractor objects. By probing objects of known geometry with the pointer, we demonstrate acceptable accuracy of 3D localization.Comment: 8 pages, 6 figures, 2018 15th Conference on Computer and Robot Visio