Francisco R. Feito

Orientation, simplicity, and inclusion test for planar polygons

Abstract This paper presents a collection of robust and simple algorithms to decide the orientation, simplicity, and inclusion of planar polygons, without solving any equation systems and without using trigonometric functions.

Boolean operations on general planar polygons

Abstract Computing boolean operations between general planar polygons is one of the fundamental problems in geometric and solid modeling. In this work we present a new algorithm to calculate intersection, union and difference, valid for general planar polygons (manifold and non-manifold, with and without holes), based on a formal representation system. This formal model is based on the concept of simplicial chain, developed by Feito and Rivero (Computers & Graphics 22(5) (1998)). Using algebraic operations between simplicial chains we can obtain any general polygon and the Boolean operations between them. The fact of that our algorithm is based on simplicial chains and their operations, reduces the study of special cases, and allows us to develop a robust and efficient algorithm to calculate the intersection between general polygons.

Inclusion test for general polyhedra

Abstract This paper presents a new algorithm which tests the inclusion of a point in a general polyhedron, manifold and non-manifold, without solving any equation system and without using trigonometric functions. The algorithm is simple and robust, and easy to apply in every case.

A new algorithm for computing Boolean operations on polygons

This paper presents a new algorithm for computing Boolean operations on polygons. These kind of operations are frequently used in the geosciences in order to get spatial information from spatial data modeled as polygons. The presented algorithm is simple and easy to understand and implement. Let n be the total number of edges of all the polygons involved in a Boolean operation and k be the number of intersections of all the polygon edges. Our algorithm computes the Boolean operation in time O((n+k)log(n)). Finally, the proposed algorithm works with concave polygons with holes, and with regions composed of polygon sets. Furthermore, it can be easily adapted to work with self-intersecting polygons.

Technical section: Point in solid strategies

Testing whether a point is inside a solid is a basic operation in computer graphics. This document presents a variety of strategies for triangle meshes, a widely used data structure in computer graphics. We discuss some issues about the capabilities of each approach depending on the situation, taking into account memory and CPU usage. A practical comparison of the performance of several strategies is also presented, with implementation issues and time tables showing the performance of each algorithm. The tests highlight the strengths and weaknesses of each approach.

Technical Section: GPU-based rendering of curved polygons using simplicial coverings

In this work, we describe a new algorithm for rendering polygons defined by cubic Bezier curve segments in current GPUs. Unlike other approaches, our algorithm has a simple preprocessing that does not require computing tessellations, and can be implemented in GPU as a geometry shader. The polygon is decomposed into a set of simplices which are individually rasterized into the stencil buffer to recreate the shape that is finally rendered in the frame buffer. Each simplex is rasterized using a fragment shader that evaluates the implicit equation of the Bezier curve to discard the pixels that fall outside it. The proposed method is simple, fast, robust and general, as it can handle curved polygons with holes, several components or self-intersections.

Fast and accurate evaluation of regularized Boolean operations on triangulated solids

In this paper we present a robust and accurate method for evaluating regularized Boolean operations on triangulated solids. It allows the exact evaluation of the regularized union, intersection, difference and symmetric difference simultaneously. Moreover, this approach is simpler than other methods, including those that provide an approximate evaluation or only a rendering of the result. It is based on a simple data structure and on the use of an octree which facilitates the division of the geometry into subsets for distribution among several threads, and accelerates the spatial queries needed during the process. This method is designed to be used in a multithreaded environment and it can also be implemented using an out-of-core approach. We also present some experimental results, and a comparison with other systems that also provide an exact evaluation of the Boolean operations.

An algorithm for determining intersection segment-polygon in 3D

Abstract In the fields related to geometric modelling, it is fundamental to count on robust. In this work we present a new algorithm for testing the intersection between a segment and a polygon in 3D space. This algorithm is robust and efficient, and it is the basis of most operations developed in geometric modelling. After, we will apply this algorithm to a classic problem in computer graphics: determining the hidden faces of a scene using the ray-casting algorithm. To solve the problem, an object-oriented approach will be used.

Geometric modelling based on simplicial chains

Abstract In this work we present a mathematical formulation for geometric modelling which may be applied in spaces of any dimension. The model can be seen as an example of a graphic object algebra [see Torres, J. C. and Clares, B., Graphics objects: a mathematical abstract model for computer graphics . Computer Graphics Forum, 1993, 12 (5), 311–328 and Feito, F. R. and Torres, J. C., Boundary representation of polyhedral heterogenous solids in the context of a graphics objects algebra. The Visual Computer , 1997, 13 (2), 64–77]. It is based on the concept of simplicial chain which is considered a particular case of the polyhedral chains presented by Whitney ( Geometric Integration Theory , Princeton University Press, Princeton, NJ, 1957). From the algebraic operations with simplicial chains we can obtain any element of the general polyhedral solids (manifold and non-manifold). Similarly, from the defined operations with chains, we can obtain the usual operations of geometric modelling (union, intersection and difference). So, the model can be considered a new scheme of representation of solids based on simplician chains. One of its uses is to obtain the usual operations of geometric modelling by means of operations with simplices (triangles, tetrahedra, etc) which are simple elements.

Semiautomatic detection of floor topology from CAD architectural drawings

A method for the semiautomatic detection of the topology of building floors represented as CAD drawings stored in vector file format is presented in this paper. This method involves the detection of walls and joint points amid walls and openings, and the search of intersection points amid walls. To give support to the wall detection process, this paper introduces the wall adjacency graph (WAG), a data structure created to detect walls from sets of planar segments contained in architectural floor plans. Wall adjacency graphs allow us to obtain a consistent and exhaustive set of walls very quickly (less than one second for real floor plans). A generalized version of the wall adjacency graph is also presented to deal with some of the limitations of the initial WAG. Algorithms for the detection of joint points and wall intersection points are presented as well, based on the analysis of the geometry from the input CAD drawings. Moreover, all this process works appropriately with both straight and circular segments. The obtained floor topology can later be used as input to generate 3D models of buildings, which are widely used on virtual cities, BIM systems and GIS.