Solving Geometric Constraints by Homotopy

International audienceNumerous methods have been proposed in order to solve geometric constraints, all of them having their own advantages and drawbacks. In this article, we propose an enhancement of the classical numerical methods, which are, up to now the only ones that apply to the general case

Tangent-ball techniques for shape processing

Shape processing defines a set of theoretical and algorithmic tools for creating, measuring and modifying digital representations of shapes.  Such tools are of paramount importance to many disciplines of computer graphics, including modeling, animation, visualization, and image processing.  Many applications of shape processing can be found in the entertainment and medical industries. In an attempt to improve upon many previous shape processing techniques, the present thesis explores the theoretical and algorithmic aspects of a difference measure, which involves fitting a ball (disk in 2D and sphere in 3D) so that it has at least one tangential contact with each shape and the ball interior is disjoint from both shapes. We propose a set of ball-based operators and discuss their properties, implementations, and applications.  We divide the group of ball-based operations into unary and binary as follows: Unary operators include: * Identifying details (sharp, salient features, constrictions) * Smoothing shapes by removing such details, replacing them by fillets and roundings * Segmentation (recognition, abstract modelization via centerline and radius variation) of tubular structures Binary operators include: * Measuring the local discrepancy between two shapes * Computing the average of two shapes * Computing point-to-point correspondence between two shapes * Computing circular trajectories between corresponding points that meet both shapes at right angles * Using these trajectories to support smooth morphing (inbetweening) * Using a curve morph to construct surfaces that interpolate between contours on consecutive slices The technical contributions of this thesis focus on the implementation of these tangent-ball operators and their usefulness in applications of shape processing. We show specific applications in the areas of animation and computer-aided medical diagnosis.  These algorithms are simple to implement, mathematically elegant, and fast to execute.Ph.D.Committee Chair: Jarek Rossignac; Committee Member: Greg Slabaugh; Committee Member: Greg Turk; Committee Member: Karen Liu; Committee Member: Maryann Simmon

Descriptive-geometrical methods in computer graphics: intersection between two surfaces of revolution by use of common spheres and common planes

U disertaciji je razmatrana problematika određivanja intersekcije dveju rotacionih površi. Ovo je izvedeno u petnaest poglavlja. U prvom poglavlju (Uvod) istaknuta je potreba da se presek dveju rotacionih površi određuje korišćenjem metoda koji u osnovi imaju deskriptivno geometrijski pristup. Takođe je dat pregled celog rada i kroz dijagrame je sistemcitizovana strategija za postcivku i rešavanje problema. U drugom poglavlju (Određivanje preseka površi u komjuterskoj grafici) izvršena je analiza postojećih metoda za određivanje međusobnog preseka dveju površi i uočeni su izvesni nedostaci ovih postupaka.U trećem poglavlju (Projiciranje) određena je dvodimenzionalna slika trodimenzionalnog objekta u slučajevima paralelnog i centralnog projiciranja. U četvrtom poglavlju (Kontura rotacione površi) određene su trodimenzionalne koordinate tačaka na konturi rotacione površi za oba slučaja projiciranja. U petom poglavlju (Pomoćne ravni) prikazani su matematički modeli za određivanje trodimenzionalnih koordinata tačaka prostorne presečne krive dveju rotacionih površi za tri slučaja međusobnog odnosa osa dveju rotacionih površi: ose su paralelne, seku se ili se mimoilaze. Za rešavanje problema korišćene su pomoćne ravni. U šestom poglavlju (Pomoćne lopte) korišćene su pomoćne lopte i za tri slučaja u odnosima između osa dveju rotacionih površi su formirani matematički modeli za određivanje prostorne presečene krive dveju površi. U sedmom poglavlju (Transformacije) izdvojene su ključne dvodimenzionalne i trodimenzionalne transformacije. U osmom poglavlju (Osnovna tela i njihovi odnosi) analizirana su osnovna geometrijska tela poput kocke, paralelopipeda, cilindra i lopte. Prikazani su načini za zadavanje ovih tela kao i procedure za definisanje međusobnog odnosa dva tela. U devetom poglavlju (Ravne krive u kompjuterskoj grafici) analizirane su najfrekventnije ravne krive u kompjuterskoj grafici koje se kao polazni objekti često koriste za generisanje površi. U desetom poglavlju (Površi u kompjuterskoj grafici) prikazani su različiti postupci za dobijanje površi u kompjuterskoj grafici. U jedanaestom poglavlju (Pregled aktuelnog softvera) prikazane su mogućnosti softvera za dobijanje površi i određivanje njihovog uzajamnog preseka. U dvanaestom poglavlju (Rezultati) prikazane su slike-rezultati koje reprezentuju dobre osobine formiranih matematičkih modela. U trinaestom poglavlju (Zaključak) analizirani su dobijeni rezultati i mogućnost daljeg istraživanja na istom ili sličnim problemima. U četrnaestom poglavlju (Literatura) navedene su knjige i naučni radovi koji su citirani i korišćeni tokom izrade disertacije. U petnaestom poglavlju (Prilog-listing programa) prikazan je listing Pascal programa.In the dissertation the problem of intersection between two surfaces of revolution has been considered. The subject has been presented in fifteen sections. In the first section (Introduction) the need for determination of intersecting curve between two surfaces of revolution by use of methods based on descriptive geometrical access has been underlined. The dissertation review has been given and the strategy for formulation and determination of problems by use of diagrams has been systematised. In the second section (Interseclions of surfaces in computer graphic) former approaches for surface-surface intersection have been analysed and several shortages have been noted. In the third section (Projection) the two-dimensional picture of three-dimensional object in cases of parallel and central projections has been determined. In the fourth section (Contour lines of surface of revolution) the three-dimensional point ’s coordinates on contour lines of surface of revolution for both projection cases have been solved. In the fifth section (Common planes) the mathematical models for determination three-dimensional point 's coordinates of space intersecting curve between two surfaces of revolution have been presented. Three different cases have been considered: axes of surfaces of revolution are parallel, axes are intersecting or pass each other. For determination of problem common planes have been used. In the sixth section (Common spheres) common spheres have been used. The mathematical models for determination intersecting space curve for three different cases in relations behveen axes of surfaces of revolution have been formed. In the sixth section (Common spheres) common spheres have been used. The mathematical models for determination intersecting space curve for three different cases in relations behveen axes of surfaces of revolution have been formed. In the seventh section (Transformations) the basic two-dimensional and three-dimensional transformations have been presenled. In the eighth section (Basic solids and their relations) the basic solids: cube, block, cylinder and sphere have been analysed. The assign solids ways and procedures for determination of relation between two solids have been analysed. In the ninth section (Plane curves in computer graphic) the most frequent plane curves in computer graphic which often used for determination of surfaces like starting objects have been analysed. In the tenth section (Surfaces in computer graphics) the different procedures for determination of surfaces in computer graphics have been presented. In the eleventh section (Pascal program review) the actual softwares possibilities for determination of surfaces and their intersections have been presented. In the twelfth section (Results) the pictures-results which represented good features of formed mathematical models have been presented. In the thirteenth section (Conclusion) the obtained results have been analysed as well as the possibilities of further investigations for both same and similar problems. In the fourteenth section (References) books and scientific papers, which have been either quoted or used in this dissertation, have been given. In the fifteenth section (Additional list-Program listing) the used Pascal program has been presented