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Publication# Modélisation géométrique itérative sous contraintes

Abstract

Iterative models are widely used today in CAD. They allow, with a limited number of parameters, to represent relatively complex forms through a subdivision algorithm. There is a wide variety of such models (Catmull-Clark, Doo-Sabin, L-Systems...). Most iterative models used in CAD can represent smooth shapes, such as polynomial or rational. The IFS model (Iterated Function System) is a mathematical model allowing to represent objects that can be smooth, in particular cases, or fractal, in more general cases. An IFS is defined by a set of geometric operators called "subdivision operators". These operators define an object iteratively, by successively applying this set of subdivision operators on a geometric base object. Classical subdivision schemes take as parameters a set of control points, that can be moved anywhere in space. These control points are the entry parameters of the subdivision algorithm, which uses predefined subdivision matrices to calculate the new points. In the IFS model, subdivision operators are not predefined, but customizable. These new parameters are graphically represented as movable points in space, like the control points. Each of these points, referred to as "subdivision point" is the image of a control point through a subdivision operator. The position of the control points allow to control the global aspect of the modelled object. Moving subdivision points affects the object at each level of subdivision, and therefore at smaller and smaller scales. The generated objects are not necessarily smooth, but are generally fractal. The constraints due to construction require some precise geometric properties of the modelled objects. As part of the wooden building, we want to achieve particular surface structures by assembly of wood panels. This requires modelling meshes composed of planar faces. We are particularly interested in modelling quadrangular mesh. We discard the solution of triangular meshes. This comes from constraints related to construction, and is more particularly due to the complexity of realizing assemblies around high valence vertices. The vertices in triangulated meshes have a valence of six, while in quadrangular meshes they have a valence of four. The development and implementation of solutions are relatively expensive in terms of the valence of the node. The valence of the nodes of a mesh has a direct influence on the geometry of faces ; the higher the valence of a vertex, the higher angles of faces around this vertex will be acute. Faces with acute angles are not desirable for a constructive application, because constructive elements have fragile parts and handling them during the implementation process is a delicate operation. We propose a method based on an iterative model that generates directly planarquadrilateral meshes. We start from a Minkowski sum of two curves. This process is rather limited, because it generates meshes only composed by parallelograms. We expand the possibilities for creating forms, working in a 4D homogeneous coordinate system, and projecting these forms in the 3D modelling space. Using projective geometry allows to extend the method by additional parameters such as the weight of points. This allows to reach a relatively general range of surface meshes.

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Related concepts (2)

Construction

Construction is a general term meaning the art and science to form objects, systems, or organizations, and comes from Latin constructio (from com- "together" and struere "to pile up") and Old French construction. To construct is the verb: the act of building, and the noun is construction: how something is built, the nature of its structure. In its most widely used context, construction covers the processes involved in delivering buildings, infrastructure, industrial facilities, and associated activities through to the end of their life.

Geometry

Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.