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**Solid modeling** (or **modelling**) is a consistent set of principles for mathematical and computer modeling of three-dimensional solids. Solid modeling is distinguished from related areas of geometric modeling and computer graphics 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.

**Geometric modeling** is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes.

**Computer graphics** are pictures and films created using computers. Usually, the term refers to computer-generated image data created with the help of specialized graphical hardware and software. It is a vast and recently developed area of computer science. The phrase was coined in 1960, by computer graphics researchers Verne Hudson and William Fetter of Boeing. It is often abbreviated as CG, though sometimes erroneously referred to as computer-generated imagery (CGI).

In 3D computer graphics, **3D modeling** is the process of developing a mathematical representation of any *surface* of an object in three dimensions via specialized software. The product is called a **3D model**. Someone who works with 3D models may be referred to as a **3D artist**. It can be displayed as a two-dimensional image through a process called *3D rendering* or used in a computer simulation of physical phenomena. The model can also be physically created using 3D printing devices.

- Overview
- Mathematical foundations
- Solid representation schemes
- Parameterized primitive instancing
- Spatial occupancy enumeration
- Cell decomposition
- Boundary representation
- Surface mesh modeling
- Constructive solid geometry
- Sweeping
- Implicit representation
- Parametric and feature-based modeling
- History of solid modelers
- Computer-aided design
- Parametric modeling
- Medical solid modeling
- Engineering
- See also
- References
- External links

The use of solid modeling techniques allows for the automation 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.

**Machining** is any of various processes in which a piece of raw material is cut into a desired final shape and size by a controlled material-removal process. The processes that have this common theme, controlled material removal, are today collectively known as **subtractive manufacturing**, in distinction from processes of controlled material addition, which are known as additive manufacturing. Exactly what the "controlled" part of the definition implies can vary, but it almost always implies the use of machine tools.

An **assembly line** is a manufacturing process in which parts are added as the semi-finished assembly moves from workstation to workstation where the parts are added in sequence until the final assembly is produced. By mechanically moving the parts to the assembly work and moving the semi-finished assembly from work station to work station, a finished product can be assembled faster and with less labor than by having workers carry parts to a stationary piece for assembly.

**Sheet metal** is metal formed by an industrial process into thin, flat pieces. Sheet metal is one of the fundamental forms used in metalworking and it can be cut and bent into a variety of shapes. Countless everyday objects are fabricated from sheet metal. Thicknesses can vary significantly; extremely thin sheets are considered foil or leaf, and pieces thicker than 6 mm (0.25 in) are considered plate steel or "structural steel."

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 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.

In topology and mathematics in general, the **boundary** of a subset *S* of a topological space *X* is the set of points which can be approached both from *S* and from the outside of *S*. More precisely, it is the set of points in the closure of *S* not belonging to the interior of *S*. An element of the boundary of *S* is called a **boundary point** of *S*. The term **boundary operation** refers to finding or taking the boundary of a set. Notations used for boundary of a set *S* include bd(*S*), fr(*S*), and ∂*S*. Some authors use the term **frontier** instead of boundary in an attempt to avoid confusion with the concept of boundary used in algebraic topology and manifold theory. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, the term frontier has been used to describe the **residue** of *S*, namely *S* \ *S*. Hausdorff named the intersection of *S* with its boundary the **border** of *S*.

In geometry, **Euclidean space** encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and similar spaces of higher dimension. It is named after the Ancient Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions.

**Algebraic topology** is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

According to the continuum point-set model of solidity, all the points of any *X* ⊂ ℝ^{3} can be classified according to their neighborhoods with respect to *X* as interior, exterior, or boundary points. Assuming ℝ^{3} is endowed with the typical Euclidean metric, a neighborhood of a point *p* ∈*X* takes the form of an open ball. For *X* to be considered solid, every neighborhood of any *p* ∈*X* 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* ⊂ ℝ^{3} 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 ℝ^{3} (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 ∪^{∗}, ∩^{∗}, and −^{∗}.

In mathematics, specifically in topology, the **interior** of a subset of a topological space is the union of all open subsets of that set. A point that is in the interior of *S* is an **interior point** of *S*.

In topology, the **exterior** of a subset *S* of a topological space *X* is the union of all open sets of *X* which are disjoint from *S*. It is itself an open set and is disjoint from *S*. The exterior of *S* is denoted by

In mathematics, a **ball** is the space bounded by a sphere. It may be a **closed ball** or an **open ball**.

The combinatorial characterization of a set *X* ⊂ ℝ^{3} 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.

In topology, a **CW complex** is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation.

In mathematical analysis and related areas of mathematics, a set is called **bounded**, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called **unbounded**. The word bounded makes no sense in a general topological space without a corresponding metric.

**Stratification** has several usages in mathematics.

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 ℝ^{n} 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−1*-dimensional polyhedron having homogeneously *n−1*-dimensional neighborhoods.

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.

This scheme is based on motion 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.

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.

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] }

In this scheme a solid is represented by the cellular decomposition of its boundary. Since the boundaries of solids have the distinguishing property that they separate space into regions defined by the interior of the solid and the complementary exterior according to the Jordan-Brouwer theorem discussed above, every point in space can unambiguously be tested against the solid by testing the point against the boundary of the solid. Recall that ability to test every point in the solid provides a guarantee of solidity. Using ray casting it is possible to count the number of intersections of a cast ray against the boundary of the solid. Even number of intersections correspond to exterior points, and odd number of intersections correspond to interior points. The assumption of boundaries as manifold cell complexes forces any boundary representation to obey disjointedness of distinct primitives, i.e. there are no self-intersections that cause non-manifold points. In particular, the manifoldness condition implies all pairs of vertices are disjoint, pairs of edges are either disjoint or intersect at one vertex, and pairs of faces are disjoint or intersect at a common edge. Several data structures that are combinatorial maps have been developed to store boundary representations of solids. In addition to planar faces, modern systems provide the ability to store quadrics and NURBS surfaces as a part of the boundary representation. Boundary representations have evolved into a ubiquitous representation scheme of solids in most commercial geometric modelers because of their flexibility in representing solids exhibiting a high level of geometric complexity.

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 (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.

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.

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 the conditions , , and represent, respectively, a plane and two open linear halfspaces. More complex functional primitives may be defined by boolean combinations of simpler predicates. Furthermore, the theory of R-functions allow conversions of such representations into a single function inequality for any closed semi analytic set. Such a representation can be converted to a boundary representation using polygonization algorithms, for example, the marching cubes algorithm.

Features are defined to be parametric shapes associated with *attributes* such as intrinsic geometric parameters (length, width, depth etc.), position and orientation, geometric tolerances, material properties, and references to other features.^{ [8] } Features also provide access to related production processes and resource models. Thus, features have a semantically higher level than primitive closed regular sets. Features are generally expected to form a basis for linking CAD with downstream manufacturing applications, and also for organizing databases for design data reuse. Parametric feature based modeling is frequently combined with constructive binary solid geometry (CSG) to fully describe systems of complex objects in engineering.

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The historical development of solid modelers has to be seen in context of the whole history of CAD, the key milestones being the development of the research system BUILD followed by its commercial spin-off Romulus which went on to influence the development of Parasolid, ACIS and Solid Modeling Solutions. One of the first CAD developers in the Commonwealth of Independent States (CIS), ASCON began internal development of its own solid modeler in the 1990s.^{ [9] } In November 2012, the mathematical division of ASCON became a separate company, and was named C3D Labs. It was assigned the task of developing the C3D geometric modeling kernel as a standalone product — the only commercial 3D modeling kernel from Russia.^{ [10] } Other contributions came from Mäntylä, with his GWB and from the GPM project which contributed, among other things, hybrid modeling techniques at the beginning of the 1980s. This is also when the Programming Language of Solid Modeling PLaSM was conceived at the University of Rome.

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The modeling of solids is only the minimum requirement of a CAD system's capabilities. Solid modelers have become commonplace in engineering departments in the last ten years^{[ when? ]} due to faster computers and competitive software pricing. Solid modeling software creates a virtual 3D representation of components for machine design and analysis.^{ [11] } A typical graphical user interface includes programmable macros, keyboard shortcuts and dynamic model manipulation. The ability to dynamically re-orient the model, in real-time shaded 3-D, is emphasized and helps the designer maintain a mental 3-D image.

A solid part model generally consists of a group of features, added one at a time, until the model is complete. Engineering solid models are built mostly with sketcher-based features; 2-D sketches that are swept along a path to become 3-D. These may be cuts, or extrusions for example. Design work on components is usually done within the context of the whole product using assembly modeling methods. An assembly model incorporates references to individual part models that comprise the product.^{ [12] }

Another type of modeling technique is 'surfacing' (Freeform surface modeling). Here, surfaces are defined, trimmed and merged, and filled to make solid. The surfaces are usually defined with datum curves in space and a variety of complex commands. Surfacing is more difficult, but better applicable to some manufacturing techniques, like injection molding. Solid models for injection molded parts usually have both surfacing and sketcher based features.

Engineering drawings can be created semi-automatically and reference the solid models.

Parametric modeling uses parameters to define a model (dimensions, for example). Examples of parameters are: dimensions used to create model features, material density, formulas to describe swept features, imported data (that describe a reference surface, for example). The parameter may be modified later, and the model will update to reflect the modification. Typically, there is a relationship between parts, assemblies, and drawings. A part consists of multiple features, and an assembly consists of multiple parts. Drawings can be made from either parts or assemblies.

Example: A shaft is created by extruding a circle 100 mm. A hub is assembled to the end of the shaft. Later, the shaft is modified to be 200 mm long (click on the shaft, select the length dimension, modify to 200). When the model is updated the shaft will be 200 mm long, the hub will relocate to the end of the shaft to which it was assembled, and the engineering drawings and mass properties will reflect all changes automatically.

Related to parameters, but slightly different, are constraints. Constraints are relationships between entities that make up a particular shape. For a window, the sides might be defined as being parallel, and of the same length. Parametric modeling is obvious and intuitive. But for the first three decades of CAD this was not the case. Modification meant re-draw, or add a new cut or protrusion on top of old ones. Dimensions on engineering drawings were *created*, instead of *shown*. Parametric modeling is very powerful, but requires more skill in model creation. A complicated model for an injection molded part may have a thousand features, and modifying an early feature may cause later features to fail. Skillfully created parametric models are easier to maintain and modify. Parametric modeling also lends itself to data re-use. A whole family of capscrews can be contained in one model, for example.

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Modern computed axial tomography and magnetic resonance imaging scanners can be used to create solid models of internal body features, so-called volume rendering. Optical 3D scanners can be used to create point clouds or polygon mesh models of external body features.

Uses of medical solid modeling;

- Visualization
- Visualization of specific body tissues (just blood vessels and tumor, for example)
- Designing prosthetics, orthotics, and other medical and dental devices (this is sometimes called mass customization)
- Creating polygon mesh models for rapid prototyping (to aid surgeons preparing for difficult surgeries, for example)
- Combining polygon mesh models with CAD solid modeling (design of hip replacement parts, for example)
- Computational analysis of complex biological processes, e.g. air flow, blood flow
- Computational simulation of new medical devices and implants
*in vivo*

If the use goes beyond visualization of the scan data, processes like image segmentation and image-based meshing will be necessary to generate an accurate and realistic geometrical description of the scan data.

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Because CAD programs running on computers “understand” the true geometry comprising complex shapes, many attributes of/for a 3‑D solid, such as its center of gravity, volume, and mass, can be quickly calculated. For instance, the cube shown at the top of this article measures 8.4 mm from flat to flat. Despite its many radii and the shallow pyramid on each of its six faces, its properties are readily calculated for the designer, as shown in the screenshot at right.

- PLaSM - Programming Language of Solid Modeling.
- Computer graphics
- Computational geometry
- Euler boundary representation
- Engineering drawing
- Technical drawing
- List of CAD companies

A **wire-frame model** is a visual presentation of a 3-dimensional (3D) or physical object used in 3D computer graphics. It is created by specifying each edge of the physical object where two mathematically continuous smooth surfaces meet, or by connecting an object's constituent vertices using straight lines or curves. The object is projected into screen space by drawing lines at the location of each edge. The term wire frame comes from designers using metal wire to represent the three-dimensional shape of solid objects. 3D wire frame allows the construction and manipulation of solids and solid surfaces. The 3D solid modeling technique efficiently draws higher quality representations of solids than the conventional line drawing.

**Computer-aided design** (**CAD**) is the use of computers to aid in the creation, modification, analysis, or optimization of a design. CAD software is used to increase the productivity of the designer, improve the quality of design, improve communications through documentation, and to create a database for manufacturing. CAD output is often in the form of electronic files for print, machining, or other manufacturing operations. The term **CADD** is also used.

**Computational geometry** is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry. While modern computational geometry is a recent development, it is one of the oldest fields of computing with history stretching back to antiquity.

The term **geometric primitive**, or **prim**, in computer graphics and CAD systems is used in various senses, with the common meaning of the simplest geometric objects that the system can handle. Sometimes the subroutines that draw the corresponding objects are called "geometric primitives" as well. The most "primitive" primitives are point and straight line segment, which were all that early vector graphics systems had.

**Constructive solid geometry** (**CSG**) is a technique used in solid modeling. Constructive solid geometry allows a modeler to create a complex surface or object by using Boolean operators to combine simpler objects. potentially generating visually complex objects by combining a few primitive ones.

A **polygon mesh** is a collection of **vertices**, **edges** and **faces** that defines the shape of a polyhedral object in 3D computer graphics and solid modeling. The faces usually consist of triangles, quadrilaterals, or other simple convex polygons, since this simplifies rendering, but may also be composed of more general concave polygons, or polygons with holes.

**BRL-CAD** is a constructive solid geometry (CSG) solid modeling computer-aided design (CAD) system. It includes an interactive geometry editor, ray tracing support for graphics rendering and geometric analysis, computer network distributed framebuffer support, scripting, image-processing and signal-processing tools. The entire package is distributed in source code and binary form.

In solid modeling and computer-aided design, **boundary representation**—often abbreviated as **B-rep** or **BREP**—is a method for representing shapes using the limits. A solid is represented as a collection of connected surface elements, the boundary between solid and non-solid.

**Mesh generation** is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells. Often these cells form a simplicial complex. Usually the cells partition the geometric input domain. Mesh cells are used as discrete local approximations of the larger domain. Meshes are created by computer algorithms, often with human guidance through a GUI, depending on the complexity of the domain and the type of mesh desired. The goal is to create a mesh that accurately captures the input domain geometry, with high-quality (well-shaped) cells, and without so many cells as to make subsequent calculations intractable. The mesh should also be fine in areas that are important for the subsequent calculations.

**Generative Modelling Language** (GML) in computer graphics and generative computer programming is a very simple programming language for the concise description of complex 3D shapes. It follows the "Generative Modelling" paradigm, where complex datasets are represented by "lists of operations" rather than by lists of objects, which is for instance the case in a relational database.

**Function Representation** is used in solid modeling, volume modeling and computer graphics. FRep was introduced in "Function representation in geometric modeling: concepts, implementation and applications" as a uniform representation of multidimensional geometric objects (shapes). An object as a point set in multidimensional space is defined by a single continuous real-valued function of point coordinates which is evaluated at the given point by a procedure traversing a tree structure with primitives in the leaves and operations in the nodes of the tree. The points with belong to the object, and the points with are outside of the object. The point set with is called an isosurface.

The term "**feature**" implies different meanings in different engineering disciplines. This has resulted in many ambiguous definitions for feature. A feature, in computer-aided design (CAD), usually refers to a region of a part with some interesting geometric or topological properties. These are more precisely called form features. Form features contain both shape information and parametric information of a region of interest. They are now ubiquitous in most current CAD software, where they are used as the primary means of creating 3D geometric models. Examples of form features are extruded boss, loft, etc. Form feature is not the only type of feature that is discussed in CAD literature. Sometimes a part's functional or manufacturing features of the subject of attention. Although it is quite possible to see form features and manufacturing features are called by the same name, they are not exactly the same concepts. For example, one may either use the name "pocket" to refer to a swept cut on the boundary of a part model, or to refer to a trace left on the part boundary by a specific machining operation. The former is exclusively concerned with a geometric shape whereas the latter is concerned with both the geometric shape and a manufacturing operation, needing more parameters in its definition. As such, a manufacturing feature can be minimally defined as a form feature, but not necessarily vice versa. Machining features are an important subset of manufacturing features. A machining feature can be regarded as the volume swept by a "cutting" tool, which is always a negative (subtracted) volume. Finally, there is also the concept of assembly feature, which encodes the assembly method between connected components.

**Computer graphics** is a sub-field of Computer Science which studies methods for digitally synthesizing and manipulating visual content. Although the term often refers to the study of three-dimensional computer graphics, it also encompasses two-dimensional graphics and image processing.

A **combinatorial map** is a combinatorial object modelling topological structures with subdivided objects. Historically, the concept was introduced informally by J. Edmonds for polyhedral surfaces which are planar graphs. It was given its first definite formal expression under the name "Constellations" by A. Jacques but the concept was already extensively used under the name "rotation" by Gerhard Ringel and J.W.T. Youngs in their famous solution of the Heawood map-coloring problem. The term "constellation" was not retained and instead "combinatorial map" was favored. The concept was later extended to represent higher-dimensional orientable subdivided objects. Combinatorial maps are used as efficient data structures in image representation and processing, in geometrical modeling. This model is related to simplicial complexes and to combinatorial topology. Note that combinatorial maps were extended to generalized maps that allow also to represent non-orientable objects like the Möbius strip and the Klein bottle. A combinatorial map is a boundary representation model; it represents object by its boundaries.

**Geometric design** (**GD**) is a branch of computational geometry. It deals with the construction and representation of free-form curves, surfaces, or volumes and is closely related to geometric modeling. Core problems are curve and surface modelling and representation. GD studies especially the construction and manipulation of curves and surfaces given by a set of points using polynomial, rational, piecewise polynomial, or piecewise rational methods. The most important instruments here are parametric curves and parametric surfaces, such as Bézier curves, spline curves and surfaces. An important non-parametric approach is the level-set method.

**Cobalt** is a parametric-based computer-aided design (CAD) and 3D modeling program that runs on both Macintosh and Microsoft Windows operating systems. The program combines the direct-modeling way to create and edit objects and the highly structured, history-driven parametric way exemplified by programs like Pro/ENGINEER. A product of Ashlar-Vellum, Cobalt is Wireframe-based and history-driven with associativity and 2D equation-driven parametrics and constraints. It offers surfacing tools, mold design tools, detailing, and engineering features. Cobalt includes a library of 149,000 mechanical parts.

**C3D Toolkit** is a geometric modeling kit originally developed by ASCON Group, now by C3D Labs, using C++ and written in Visual Studio. C3D Toolkit responsible for constructing and editing geometric models. It can be licensed by other companies for use in their 3D computer graphics software products. The most widely known software in which C3D Toolkit is typically used are computer aided design (CAD), computer-aided manufacturing (CAM), and computer-aided engineering (CAE) systems.

- 1 2 Shapiro, Vadim (2001).
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*ACM Computing Surveys*. ACM Computing Surveys.**12**(4): 437–464. doi:10.1145/356827.356833. - ↑ Hatcher, A. (2002).
*Algebraic Topology*. Cambridge University Press. Retrieved 20 April 2010. - ↑ Canny, John F. (1987).
*The Complexity of Robot Motion Planning*. MIT press, ACM doctoral dissertation award. Retrieved 20 April 2010. - ↑ Ziegler, M. (2004). "Computable Operators on Regular Sets". Wiley. doi:10.1002/malq.200310107.
- ↑ Mantyla, M., Nau, D. , and Shah, J. (1996). "Challenges in feature based manufacturing research".
*Communications of the ACM*. Communications of the ACM.**39**(2): 77–85. doi:10.1145/230798.230808.CS1 maint: Multiple names: authors list (link) - ↑ Yares, Evan (April 2013). "Russian CAD".
*Design World*. WTWH Media, LLC.**8**(4). ISSN 1941-7217. Archived from the original on 30 January 2015. - ↑ Golovanov, Nikolay (2014).
*Geometric Modeling: The mathematics of shapes*. CreateSpace Independent Publishing Platform (December 24, 2014). p. Back cover. ISBN 978-1497473195. - ↑ LaCourse, Donald (1995). "2".
*Handbook of Solid Modeling*. McGraw Hill. p. 2.5. ISBN 978-0-07-035788-4. - ↑ LaCourse, Donald (1995). "11".
*Handbook of Solid Modeling*. McGraw Hill. p. 111.2. ISBN 978-0-07-035788-4.

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