A New Algorithm for Boolean Operations on General Polygons

International audienceA new algorithm for Boolean operations on general planar polygons is presented. It is available for general planar polygons (manifold or non-manifold, with or without holes). Edges of the two general polygons are subdivided at the intersection points and touching points. Thus, the boundaryof the Boolean operation resultant polygon is made of some whole edges of the polygons after the subdivision process. We use the simplex theory to build the basic mathematical model of the new algorithm. The subordination problem between an edge and a polygon is reduced to a problem of determining whether a point is on some edges of some simplices or inside the simplices, and the associated simplicial chain of the resultant polygon is just an assembly of some simplices and their coefficients of the two polygons after the subdivision process. Examples show that the running time required bythe new algorithm is less than one-third of that bythe Rivero and Feito algorithm

Analysis of General Polygon Boolean Operation Algorithms

Tato diplomová práce se zabývá algoritmy pro booleovské operace nad obecnými polygony. Mezi booleovské operace se řadí např. průnik, sjednocení nebo rozdíl. Obecný polygon může být např. sebeprotínající s otvorem. Pravděpodobně nejznámější booleovskou operací je oříznutí polygonu obdélníkovým oknem. Na začátku práce jsou vysvětleny základní pojmy. V další části je popsán princip vybrané množiny algoritmů, které provádějí booleovské operace nad polygony. V závěrečné části je provedeno komplexní srovnání implementací algoritmů jak z hlediska rychlosti výpočtu, tak z hlediska schopnosti zpracovat různě složité typy vstupních dat. Výstupem práce je souhrnné zhodnocení všech algoritmů a dynamická knihovna, která obsahuje implementace všech algoritmů.This thesis deals with general polygon boolean operation algorithms. Boolean operations are e.g. intersection, union or difference. A general polygon can be e.g. a selfinterecting polygon with inner hole. Clipping of polygons against a rectangular window is probably the most familiar boolean operation on polygons. At first, basic definitions are listed. Then the principles of a selected set of boolean operation algorithms are reviewed. Finally, a complex comparison of the algorithms is undertaken. Performance as well as the ability to handle degenerate cases are tested. The output of this thesis is an overall evaluation of algorithm properties and a dynamic library that contains the implementation of all of the tested algorithms.

A new 3D modelling paradigm for discrete model

Until few years ago, 3D modelling was a topic confined into a professional environment. Nowadays technological innovations, the 3D printer among all, have attracted novice users to this application field. This sudden breakthrough was not supported by adequate software solutions. The 3D editing tools currently available do not assist the non-expert user during the various stages of generation, interaction and manipulation of 3D virtual models. This is mainly due to the current paradigm that is largely supported by two-dimensional input/output devices and strongly affected by obvious geometrical constraints. We have identified three main phases that characterize the creation and management of 3D virtual models. We investigated these directions evaluating and simplifying the classic editing techniques in order to propose more natural and intuitive tools in a pure 3D modelling environment. In particular, we focused on freehand sketch-based modelling to create 3D virtual models, interaction and navigation in a 3D modelling environment and advanced editing tools for free-form deformation and objects composition. To pursuing these goals we wondered how new gesture-based interaction technologies can be successfully employed in a 3D modelling environments, how we could improve the depth perception and the interaction in 3D environments and which operations could be developed to simplify the classical virtual models editing paradigm. Our main aims were to propose a set of solutions with which a common user can realize an idea in a 3D virtual model, drawing in the air just as he would on paper. Moreover, we tried to use gestures and mid-air movements to explore and interact in 3D virtual environment, and we studied simple and effective 3D form transformations. The work was carried out adopting the discrete representation of the models, thanks to its intuitiveness, but especially because it is full of open challenges

Technical section A new algorithm for Boolean operations on general polygons

A new algorithm for Boolean operations on general planar polygons is presented. It is available for general planar polygons (manifold or non-manifold, with or without holes). Edges of the two general polygons are subdivided at the intersection points and touching points. Thus, the boundaryof the Boolean operation resultant polygon is made of some whole edges of the polygons after the subdivision process. We use the simplex theory to build the basic mathematical model of the new algorithm. The subordination problem between an edge and a polygon is reduced to a problem of determining whether a point is on some edges of some simplices or inside the simplices, and the associated simplicial chain of the resultant polygon is just an assembly of some simplices and their coefficients of the two polygons after the subdivision process. Examples show that the running time required bythe new algorithm is less than one-third of that bythe Rivero and Feito algorithm