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 simple construction method for sequentially tidying up 2D online freehand sketches

This paper presents a novel constructive approach to sequentially tidying up 2D online freehand sketches for further 3D interpretation in a conceptual design system. Upon receiving a sketch stroke, the system first identifies it as a 2D primitive and then automatically infers its 2D geometric constraints related to previous 2D geometry (if any). Based on recognized 2D constraints, the identified geometry will be modified accordingly to meet its constraints. The modification is realized in one or two sequent geometric constructions in consistence with its degrees of freedom. This method can produce 2D configurations without iterative procedures to solve constraint equations. It is simple and easy to use for a real-time application. Several examples are tested and discussed

Equations, inequations and inequalities characterizing the configurations of two real projective conics

Couples of proper, non-empty real projective conics can be classified modulo rigid isotopy and ambient isotopy. We characterize the classes by equations, inequations and inequalities in the coefficients of the quadratic forms defining the conics. The results are well--adapted to the study of the relative position of two conics defined by equations depending on parameters.Comment: 31 pages. See also http://emmanuel.jean.briand.free.fr/publications/twoconics/ Added references to important prior work on the subject. The title changed accordingly. Some typos and imprecisions corrected. To be published in Applicable Algebra in Engineering, Communication and Computin

On the capture and representation of fonts

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The commercial need to capture, process and represent the shape and form of an outline has lead to the development of a number of spline routines. These use a mathematical curve format that approximates the contours of a given shape. The modelled outline lends itself to be used on, and for, a variety of purposes. These include graphic screens, laser printers and numerically controlled machines. The latter can be employed for cutting foil, metal. plastic and stone. One of the most widely used software design packages has been the lKARUS system. This, developed by URW of Hamburg (Gennany), employs a number of mathematical descriptions that facilitate the process of both modelling and representation of font characters. It uses a variety of curve formats, including Bezier cubics, general conics and parabolics. The work reported in this dissertation focuses on developing improved techniques, primarily. for the lKARUS system. This includes two algorithms which allow a Bezier cubic description, two for a general conic representation and, yet another, two for the parabolic case. In addition, a number of algorithms are presented which promote conversions between these mathematical forms; for example, Bezier cubics to a general conic form. Furthennore, algorithms are developed to assist the process of rasterising both cubic and quadratic arcs.This study was partly funded by the Science and Education Research Council (SERC)

On the works of Euler and his followers on spherical geometry

We review and comment on some works of Euler and his followers on spherical geometry. We start by presenting some memoirs of Euler on spherical trigonometry. We comment on Euler's use of the methods of the calculus of variations in spherical trigonometry. We then survey a series of geometrical resuls, where the stress is on the analogy between the results in spherical geometry and the corresponding results in Euclidean geometry. We elaborate on two such results. The first one, known as Lexell's Theorem (Lexell was a student of Euler), concerns the locus of the vertices of a spherical triangle with a fixed area and a given base. This is the spherical counterpart of a result in Euclid's Elements, but it is much more difficult to prove than its Euclidean analogue. The second result, due to Euler, is the spherical analogue of a generalization of a theorem of Pappus (Proposition 117 of Book VII of the Collection) on the construction of a triangle inscribed in a circle whose sides are contained in three lines that pass through three given points. Both results have many ramifications, involving several mathematicians, and we mention some of these developments. We also comment on three papers of Euler on projections of the sphere on the Euclidean plane that are related with the art of drawing geographical maps.Comment: To appear in Ganita Bharati (Indian Mathematics), the Bulletin of the Indian Society for History of Mathematic

Tolerance Zone-Based Grouping Method for Online Multiple Overtracing Freehand Sketches

Multiple overtracing strokes are common drawing behaviors in freehand sketching; that is, additional strokes are often drawn repeatedly over the existing ones to add more details. This paper proposes a method based on stroke-tolerance zones to group multiple overtraced strokes which are drawn to express a 2D primitive, aiming to convert online freehand sketches into 2D line drawings, which is a base for further 3D reconstruction. Firstly, after the user inputs a new stroke, a tolerance zone around the stroke is constructed by reference to its polygonal approximation points obtained from the stroke preprocessing. Then, the input strokes are divided into stroke groups, each representing a primitive through the stroke grouping process based on the overtraced ratio of two strokes. At last, each stroke group is fitted into one or more 2D geometric primitives including line segments, polylines, ellipses, and arcs. The proposed method groups two strokes together based on their screen-space proximity directly instead of classifying and fitting them firstly, so that it can group strokes of arbitrary shapes. A sketch-recognition prototype system has been implemented to test the effectiveness of the proposed method. The results showed that the proposed method could support online multiple overtracing freehand sketching with no limitation on drawing sequence, but it only deals with strokes with relatively high overtraced ratio