[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/solid-modeling-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki21\/solid-modeling-wikipedia\/","headline":"Solid modeling – Wikipedia","name":"Solid modeling – Wikipedia","description":"before-content-x4 Set of principles for modeling solid geometry The geometry in solid modeling is fully described in 3\u2011D space; objects","datePublished":"2014-08-25","dateModified":"2014-08-25","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki21\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/0\/0f\/Jack-in-cube_solid_model%2C_light_background.gif","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/0\/0f\/Jack-in-cube_solid_model%2C_light_background.gif","height":"274","width":"180"},"url":"https:\/\/wiki.edu.vn\/en\/wiki21\/solid-modeling-wikipedia\/","wordCount":6510,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4Set of principles for modeling solid geometry  The geometry in solid modeling is fully described in 3\u2011D space; objects can be viewed from any angle.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Solid modeling (or solid modelling) is a consistent set of principles for mathematical and computer modeling of three-dimensional shapes (solids). Solid modeling is distinguished from related areas of geometric modeling and computer graphics, such as 3D modeling, by its emphasis on physical fidelity.[1] Together, the principles of geometric and solid modeling form the foundation of 3D-computer-aided design and in general support the creation, exchange, visualization, animation, interrogation, and annotation of digital models of physical objects.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsOverview[edit]Mathematical foundations[edit]Solid representation schemes[edit]Primitive instancing[edit]Spatial occupancy enumeration[edit]Cell decomposition[edit]Surface mesh modeling[edit]Constructive solid geometry[edit]Sweeping[edit]Implicit representation[edit]Parametric and feature-based modeling[edit]History of solid modelers[edit]Computer-aided design[edit]Parametric modeling[edit]Medical solid modeling[edit]Engineering[edit]See also[edit]References[edit]External links[edit]Overview[edit]The use of solid modeling techniques allows for the automation process of several difficult engineering calculations that are carried out as a part of the design process. Simulation, planning, and verification of processes such as machining and assembly were one of the main catalysts for the development of solid modeling. More recently, the range of supported manufacturing applications has been greatly expanded to include sheet metal manufacturing, injection molding, welding, pipe routing, etc. Beyond traditional manufacturing, solid modeling techniques serve as the foundation for rapid prototyping, digital data archival and reverse engineering by reconstructing solids from sampled points on physical objects, mechanical analysis using finite elements, motion planning and NC path verification, kinematic and dynamic analysis of mechanisms, and so on. A central problem in all these applications is the ability to effectively represent and manipulate three-dimensional geometry in a fashion that is consistent with the physical behavior of real artifacts. Solid modeling research and development has effectively addressed many of these issues, and continues to be a central focus of computer-aided engineering.Mathematical foundations[edit]The notion of solid modeling as practised today relies on the specific need for informational completeness in mechanical geometric modeling systems, in the sense that any computer model should support all geometric queries that may be asked of its corresponding physical object. The requirement implicitly recognizes the possibility of several computer representations of the same physical object as long as any two such representations are consistent. It is impossible to computationally verify informational completeness of a representation unless the notion of a physical object is defined in terms of computable mathematical properties and independent of any particular representation. Such reasoning led to the development of the modeling paradigm that has shaped the field of solid modeling as we know it today.[2]All manufactured components have finite size and well behaved boundaries, so initially the focus was on mathematically modeling rigid parts made of homogeneous isotropic material that could be added or removed. These postulated properties can be translated into properties of regions, subsets of three-dimensional Euclidean space. The two common approaches to define “solidity” rely on point-set topology and algebraic topology respectively. Both models specify how solids can be built from simple pieces or cells.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4  Regularization of a 2D set by taking the closure of its interiorAccording to the continuum point-set model of solidity, all the points of any X \u2282 \u211d3 can be classified according to their neighborhoods with respect to X as interior, exterior, or boundary points. Assuming \u211d3 is endowed with the typical Euclidean metric, a neighborhood of a point p \u2208X takes the form of an open ball. For X to be considered solid, every neighborhood of any p \u2208X must be consistently three dimensional; points with lower-dimensional neighborhoods indicate a lack of solidity. Dimensional homogeneity of neighborhoods is guaranteed for the class of closed regular sets, defined as sets equal to the closure of their interior. Any X \u2282 \u211d3 can be turned into a closed regular set or “regularized” by taking the closure of its interior, and thus the modeling space of solids is mathematically defined to be the space of closed regular subsets of \u211d3 (by the Heine-Borel theorem it is implied that all solids are compact sets). In addition, solids are required to be closed under the Boolean operations of set union, intersection, and difference (to guarantee solidity after material addition and removal). Applying the standard Boolean operations to closed regular sets may not produce a closed regular set, but this problem can be solved by regularizing the result of applying the standard Boolean operations.[3] The regularized set operations are denoted \u222a\u2217, \u2229\u2217, and \u2212\u2217.The combinatorial characterization of a set X \u2282 \u211d3 as a solid involves representing X as an orientable cell complex so that the cells provide finite spatial addresses for points in an otherwise innumerable continuum.[1] The class of semi-analytic bounded subsets of Euclidean space is closed under Boolean operations (standard and regularized) and exhibits the additional property that every semi-analytic set can be stratified into a collection of disjoint cells of dimensions 0,1,2,3. A triangulation of a semi-analytic set into a collection of points, line segments, triangular faces, and tetrahedral elements is an example of a stratification that is commonly used. The combinatorial model of solidity is then summarized by saying that in addition to being semi-analytic bounded subsets, solids are three-dimensional topological polyhedra, specifically three-dimensional orientable manifolds with boundary.[4] In particular this implies the Euler characteristic of the combinatorial boundary[5] of the polyhedron is 2. The combinatorial manifold model of solidity also guarantees the boundary of a solid separates space into exactly two components as a consequence of the Jordan-Brouwer theorem, thus eliminating sets with non-manifold neighborhoods that are deemed impossible to manufacture.The point-set and combinatorial models of solids are entirely consistent with each other, can be used interchangeably, relying on continuum or combinatorial properties as needed, and can be extended to n dimensions. The key property that facilitates this consistency is that the class of closed regular subsets of \u211dn coincides precisely with homogeneously n-dimensional topological polyhedra. Therefore, every n-dimensional solid may be unambiguously represented by its boundary and the boundary has the combinatorial structure of an n\u22121-dimensional polyhedron having homogeneously n\u22121-dimensional neighborhoods.Solid representation schemes[edit]Based on assumed mathematical properties, any scheme of representing solids is a method for capturing information about the class of semi-analytic subsets of Euclidean space. This means all representations are different ways of organizing the same geometric and topological data in the form of a data structure. All representation schemes are organized in terms of a finite number of operations on a set of primitives. Therefore, the modeling space of any particular representation is finite, and any single representation scheme may not completely suffice to represent all types of solids. For example, solids defined via combinations of regularized boolean operations cannot necessarily be represented as the sweep of a primitive moving according to a space trajectory, except in very simple cases. This forces modern geometric modeling systems to maintain several representation schemes of solids and also facilitate efficient conversion between representation schemes.Below is a list of common techniques used to create or represent solid models.[4] Modern modeling software may use a combination of these schemes to represent a solid.Primitive instancing[edit]This scheme is based on notion of families of object, each member of a family distinguishable from the other by a few parameters. Each object family is called a generic primitive, and individual objects within a family are called primitive instances. For example, a family of bolts is a generic primitive, and a single bolt specified by a particular set of parameters is a primitive instance. The distinguishing characteristic of pure parameterized instancing schemes is the lack of means for combining instances to create new structures which represent new and more complex objects. The other main drawback of this scheme is the difficulty of writing algorithms for computing properties of represented solids. A considerable amount of family-specific information must be built into the algorithms and therefore each generic primitive must be treated as a special case, allowing no uniform overall treatment.Spatial occupancy enumeration[edit]This scheme is essentially a list of spatial cells occupied by the solid. The cells, also called voxels are cubes of a fixed size and are arranged in a fixed spatial grid (other polyhedral arrangements are also possible but cubes are the simplest). Each cell may be represented by the coordinates of a single point, such as the cell’s centroid. Usually a specific scanning order is imposed and the corresponding ordered set of coordinates is called a spatial array. Spatial arrays are unambiguous and unique solid representations but are too verbose for use as ‘master’ or definitional representations. They can, however, represent coarse approximations of parts and can be used to improve the performance of geometric algorithms, especially when used in conjunction with other representations such as constructive solid geometry.Cell decomposition[edit]This scheme follows from the combinatoric (algebraic topological) descriptions of solids detailed above. A solid can be represented by its decomposition into several cells. Spatial occupancy enumeration schemes are a particular case of cell decompositions where all the cells are cubical and lie in a regular grid. Cell decompositions provide convenient ways for computing certain topological properties of solids such as its connectedness (number of pieces) and genus (number of holes). Cell decompositions in the form of triangulations are the representations used in 3d finite elements for the numerical solution of partial differential equations. Other cell decompositions such as a Whitney regular stratification or Morse decompositions may be used for applications in robot motion planning.[6]Surface mesh modeling[edit]Similar to boundary representation, the surface of the object is represented.  However, rather than complex data structures and NURBS, a simple surface mesh of vertices and edges is used.  Surface meshes can be structured (as in triangular meshes in STL files or quad meshes with horizontal and vertical rings of quadrilaterals), or unstructured meshes with randomly grouped triangles and higher level polygons.Constructive solid geometry[edit]Constructive solid geometry (CSG) is a family of schemes for representing rigid solids as Boolean constructions or combinations of primitives via the regularized set operations discussed above. CSG and boundary representations are currently the most important representation schemes for solids. CSG representations take the form of ordered binary trees where non-terminal nodes represent either rigid transformations (orientation preserving isometries) or regularized set operations. Terminal nodes are primitive leaves that represent closed regular sets. The semantics of CSG representations is clear. Each subtree represents a set resulting from applying the indicated transformations\/regularized set operations on the set represented by the primitive leaves of the subtree. CSG representations are particularly useful for capturing design intent in the form of features corresponding to material addition or removal (bosses, holes, pockets etc.). The attractive properties of CSG include conciseness, guaranteed validity of solids, computationally convenient Boolean algebraic properties, and natural control of a solid’s shape in terms of high level parameters defining the solid’s primitives and their positions and orientations. The relatively simple data structure and elegant recursive algorithms[7] have further contributed to the popularity of CSG.Sweeping[edit]The basic notion embodied in sweeping schemes is simple. A set moving through space may trace or sweep out volume (a solid) that may be represented by the moving set and its trajectory. Such a representation is important in the context of applications such as detecting the material removed from a cutter as it moves along a specified trajectory, computing dynamic interference of two solids undergoing relative motion, motion planning, and even in computer graphics applications such as tracing the motions of a brush moved on a canvas. Most commercial CAD systems provide (limited) functionality for constructing swept solids mostly in the form of a two dimensional cross section moving on a space trajectory transversal to the section. However, current research has shown several approximations of three dimensional shapes moving across one parameter, and even multi-parameter motions.Implicit representation[edit]A very general method of defining a set of points X is to specify a predicate that can be evaluated at any point in space. In other words, X is defined implicitly to consist of all the points that satisfy the condition specified by the predicate. The simplest form of a predicate is the condition on the sign of a real valued function resulting in the familiar representation of sets by equalities and inequalities. For example, if  f=ax+by+cz+d{displaystyle f=ax+by+cz+d} the conditions f(p)=0{displaystyle f(p)=0},  0″\/>, and f(p)     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki21\/solid-modeling-wikipedia\/#breadcrumbitem","name":"Solid modeling – Wikipedia"}}]}]