TheInfoList

Solid modeling (or modelling) is a consistent set of principles for mathematical and computer modeling of three-dimensional
solids Solid is one of the four fundamental states of matter 4 (four) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is an ...
. Solid modeling is distinguished from related areas of
geometric modeling __NOTOC__ Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes. The shapes studied in geometric modeling are mostly two- or three-dimensiona ...
and
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great dea ...

, such as 3D modeling, by its emphasis on physical fidelity. Together, the principles of geometric and solid modeling form the foundation of 3D-
computer-aided design Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve com ...
and in general support the creation, exchange, visualization, animation, interrogation, and annotation of digital models of physical objects.

# Overview

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 Machining is a process in which a material (often metal) is cut to a desired final shape and size by a controlled material-removal process. The processes that have this common theme are collectively called subtractive manufacturing, in contrast to ...
and
assembly Assembly may refer to: Organisations and meetings * Deliberative assembly A deliberative assembly is a gathering of members (of any kind of collective) who use parliamentary procedure Parliamentary procedure is the body of ethics, Procedural l ...

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 Sheet metal is metal A metal (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
manufacturing Manufacturing is the creation or Production (economics), production of goods with the help of equipment, Work (human activity), labor, machines, tools, and chemical or biological processing or formulation. It is the essence of secondary sector ...
,
injection molding Injection moulding (U.S. spelling: injection molding) is a manufacturing process for producing parts by injecting molten material into a Molding (process), mould, or mold. Injection moulding can be performed with a host of materials mainly inc ...

,
welding Welding is a process that joins materials, usually s or s, by using high to melt the parts together and allowing them to cool, causing . Welding is distinct from lower temperature metal-joining techniques such as and , which do not the base ...

,
pipe Pipe(s) or PIPE(S) may refer to: Common uses * Pipe (fluid conveyance) A pipe is a tubular section or hollow Cylinder (geometry), cylinder, usually but not necessarily of circle, circular cross section (geometry), cross-section, used m ...

routing, etc. Beyond traditional manufacturing, solid modeling techniques serve as the foundation for
rapid prototyping Rapid prototyping is a group of techniques used to quickly fabricate a scale model Scale or scales may refer to: Mathematics * Scale (descriptive set theory) In the mathematical discipline of descriptive set theory, a scale is a certain kind ...
, digital data archival and
reverse engineering Reverse engineering (also known as backwards engineering or back engineering) is a process or method through the application of which one attempts to understand through deductive reasoning Deductive reasoning, also deductive logic, is the process ...

by reconstructing solids from sampled points on physical objects, mechanical analysis using
finite elements The finite element method (FEM) is a widely used method for numerically solving differential equation, differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the tr ...
,
motion planning Motion planning, also path planning (also known as the navigation problem or the piano mover's problem) is a computational problem In theoretical computer science An artistic representation of a Turing machine. Turing machines are used to mode ...
and NC path verification,
kinematic Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which an object changes its ...

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 Computer-aided engineering (CAE) is the broad usage of computer software Software is a collection of Instruction (computer science), instructions and data (computing), data that tell a computer how to work. This is in contrast to Computer har ...
.

# Mathematical foundations

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. All manufactured components have finite size and well behaved boundaries, so initially the focus was on mathematically modeling rigid parts made of homogeneous
isotropic Isotropy is uniformity in all orientations; it is derived from the Greek ''isos'' (ἴσος, "equal") and ''tropos'' (τρόπος, "way"). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by ...
material that could be added or removed. These postulated properties can be translated into properties of subsets of three-dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
. The two common approaches to define solidity rely on
point-set topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
and
algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
respectively. Both models specify how solids can be built from simple pieces or cells. According to the continuum point-set model of solidity, all the points of any ''X'' ⊂ ℝ3 can be classified according to their
neighborhoods A neighbourhood (British English British English (BrE) is the standard dialect of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval E ...
with respect to ''X'' as
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
,
exterior In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric obje ...
, or
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
points. Assuming ℝ3 is endowed with the typical
Euclidean metric In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, a neighborhood of a point ''p'' ∈''X'' takes the form of an
open ball In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. For ''X'' to be considered solid, every neighborhood of any ''p'' ∈''X'' must be consistently three dimensional; points with lower-dimensional neighborhoods Gopi 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'' ⊂ ℝ3 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 ℝ3 (by the Heine-Borel theorem it is implied that all solids are
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
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. The regularized set operations are denoted ∪, ∩, and −. The combinatorial characterization of a set ''X'' ⊂ ℝ3 as a solid involves representing ''X'' as an orientable
cell complexA CW complex is a kind of a topological space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, c ...
so that the cells provide finite spatial addresses for points in an otherwise innumerable continuum. 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 Stratification may refer to: In mathematics: * Stratification (mathematics), any consistent assignment of numbers to predicate symbols * Stratified sampling , Data stratification in statistics In earth sciences: * Stable and unstable stratificati ...
into a collection of disjoint cells of dimensions 0,1,2,3. A
triangulation In trigonometry Trigonometry (from Greek '' trigōnon'', "triangle" and '' metron'', "measure") is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathe ...
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. In particular this implies the
Euler characteristic #REDIRECT Euler characteristic#REDIRECT Euler characteristic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry) ...
of the combinatorial boundary 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 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 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.

# Solid representation schemes

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 In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of ...

. 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. Modern modeling software may use a combination of these schemes to represent a solid.

## Primitive instancing

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
algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

s 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

This scheme is essentially a list of spatial ''cells'' occupied by the solid. The cells, also called
voxel In 3D computer graphics 3D computer graphics, sometimes called CGI, 3DCG or three-dimensional computer graphics (in contrast to 2D computer graphics 2D computer graphics is the Computer-generated imagery, computer-based generation of digit ...

s 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 Constructive solid geometry (CSG; formerly called computational binary solid geometry) is a technique used in solid modeling Solid modeling (or modelling) is a consistent set of principles for mathematical and computer modeling of three-dimensio ...
.

## Cell decomposition

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 propertiesIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
of solids such as its
connectedness In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
(number of pieces) and
genus Genus /ˈdʒiː.nəs/ (plural genera /ˈdʒen.ər.ə/) is a taxonomic rank In biological classification In biology, taxonomy () is the scientific study of naming, defining (Circumscription (taxonomy), circumscribing) and classifying gr ...
(number of holes). Cell decompositions in the form of triangulations are the representations used in 3d
finite elements The finite element method (FEM) is a widely used method for numerically solving differential equation, differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the tr ...
for the numerical solution of partial differential equations. Other cell decompositions such as a Whitney regular
stratification Stratification may refer to: In mathematics: * Stratification (mathematics), any consistent assignment of numbers to predicate symbols * Stratified sampling , Data stratification in statistics In earth sciences: * Stable and unstable stratificati ...
or Morse decompositions may be used for applications in robot motion planning.

## Surface mesh modeling

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

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 tree In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , ...

s where non-terminal
nodes In general, a node is a localized swelling (a "knot") or a point of intersection (a Vertex (graph theory), vertex). Node may refer to: In mathematics *Vertex (graph theory), a vertex in a mathematical graph *Node (autonomous system), behaviour fo ...
represent either rigid transformations (
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building design ...
preserving
isometries In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
) 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 Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics Linguistics is the science, scientific study of language. It e ...

algorithms have further contributed to the popularity of CSG.

## Sweeping

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

A very general method of defining a set of points ''X'' is to specify a
predicate Predicate or predication may refer to: Computer science *Syntactic predicate (in parser technology) guidelines the parser process Linguistics *Predicate (grammar), a grammatical component of a sentence Philosophy and logic * Predication (philo ...
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 $f= ax + by + cz + d$ the conditions $f\left(p\right) =0$, $f\left(p\right) > 0$, and $f\left(p\right) < 0$ 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 Marching cubes is a computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and m ...
algorithm.

## Parametric and feature-based modeling

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 A material's property (or material property) is an intensive propertyIn grammar, an intensive word form is one which denotes stronger, more forceful, or more concentrated action relative to the root on which the intensive is built. Intensives are us ...

, and references to other features. 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
database In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and sof ...

s 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.

# History of solid modelers

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 Romulus () was the legendary founder Founder or Founders may refer to: Places *Founders Park, a stadium in South Carolina, formerly known as Carolina Stadium * Founders Park, a waterside park in Islamorada, Florida#In popular culture, Islamora ...
which went on to influence the development of
Parasolid Parasolid is a geometric modeling kernelA geometric modeling kernel is a 3D solid modeling software component used in computer-aided design Computer-aided design (CAD) is the use of computer A computer is a machine that can be programme ...
,
ACIS The 3D ACIS Modeler (ACIS) is a geometric modeling kernelA geometric modeling kernel is a 3D solid modeling software component used in computer-aided design Computer-aided design (CAD) is the use of computer A computer is a machine th ...

and Solid Modeling Solutions. One of the first CAD developers in the
Commonwealth of Independent States The Commonwealth of Independent States (CIS; russian: Содружество Независимых Государств, СНГ, translit=Sodruzhestvo Nezavisimykh Gosudarstv, SNG) is a regional intergovernmental organization in Eastern Euro ...

(CIS), ASCON began internal development of its own solid modeler in the 1990s. 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 A geometric modeling __NOTOC__ Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes. The shapes studied in geometric modeling are mostly two ...
as a standalone product – the only commercial 3D modeling kernel from Russia. 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.

# Computer-aided design

The modeling of solids is only the minimum requirement of a . Solid modelers have become commonplace in engineering departments in the last ten years due to faster computers and competitive software pricing. Solid modeling software creates a virtual 3D representation of components for machine design and analysis. A typical
graphical user interface The graphical user interface (GUI "UI" by itself is still usually pronounced . or ) is a form of user interface In the industrial design Industrial design is a process of design A design is a plan or specification for the construction ...

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. 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 drawing An engineering drawing is a type of technical drawing Technical drawing, drafting or drawing, is the act and discipline Discipline is action ACTION is a bus operator in Canberra Canberra ( ) is the capital city of Austra ...
s can be created semi-automatically and reference the solid models.

## Parametric modeling

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 Injection moulding (U.S. spelling: injection molding) is a manufacturing Manufacturing is the Production (economics), production of goods through the use of Work (human activity), labor, machines, tools, and chemical or biological processing ...

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.

## Medical solid modeling

Modern
computed axial tomography A CT scan or computed tomography scan (formerly known as computed axial tomography or CAT scan) is a medical image, imaging Scientific technique, technique used in radiology to get detailed images of the body noninvasively for Diagnosis, diagnost ...
and
magnetic resonance imaging Magnetic resonance imaging (MRI) is a medical imaging Medical imaging is the technique and process of imaging Imaging is the representation or reproduction of an object's form; especially a visual representation (i.e., the formation of a ...
scanners can be used to create solid models of internal body features called
voxel In 3D computer graphics 3D computer graphics, sometimes called CGI, 3DCG or three-dimensional computer graphics (in contrast to 2D computer graphics 2D computer graphics is the Computer-generated imagery, computer-based generation of digit ...

-based models, with images generated using
volume rendering In scientific visualization Surface rendering of '' confocal_microscope..html" ;"title="onfocal_microscopy.html" ;"title="pollen.html" ;"title="Arabidopsis thaliana'' Arabidopsis_thaliana''_pollen_grains_with_Confocal_microscopy">confocal ...
. Optical
3D scanners 3D scanning is the process of analyzing a real-world object or environment to collect data on its shape and possibly its appearance (e.g. colour). The collected data can then be used to construct digital 3D modelling, 3D models. A 3D scanner can ...
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 In medicine Medicine is the science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity, awareness, or understanding of someone or something, such as facts ( descriptive knowle ...
,
orthotics Orthotics ( el, Ορθός, translit=ortho, lit=to straighten, to align) is a medical specialty A medical specialty is a branch of medical practice that is focused on a defined group of patients, diseases, skills, or philosophy. Examples inclu ...

, and other medical and dental devices (this is sometimes called
mass customization Mass customization, in marketing, manufacturing, Call centre, call centres, and management, is the use of flexible computer-aided manufacturing systems to produce custom output. Such systems combine the low unit costs of mass production processes ...
) * Creating
polygon mesh In 3D computer graphics and solid modeling, a polygon mesh is a collection of , s and s that defines the shape of a polyhedron, polyhedral object. The faces usually consist of triangles (triangle mesh), quadrilaterals (quads), or other simple c ...
models for
rapid prototyping Rapid prototyping is a group of techniques used to quickly fabricate a scale model Scale or scales may refer to: Mathematics * Scale (descriptive set theory) In the mathematical discipline of descriptive set theory, a scale is a certain kind ...
(to aid surgeons preparing for difficult surgeries, for example) * Combining polygon mesh models with
CAD Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve comm ...
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 In digital image processing Digital image processing is the use of a digital computer A computer is a machine A machine is a man-made device that uses power to apply forces and control movement to perform an action. Machines can be ...

and
image-based meshingImage-based meshing is the automated process of creating computer models for computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve pro ...
will be necessary to generate an accurate and realistic geometrical description of the scan data.

## Engineering

Because CAD programs running on computers "understand" the true geometry comprising complex shapes, many attributes of/for a 3D solid, such as its center of gravity, volume, and mass, can be quickly calculated. For instance, the cube with rounded edges 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.

*
Computational geometry 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 ar ...
*
Computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great dea ...

*
Engineering drawing An engineering drawing is a type of technical drawing Technical drawing, drafting or drawing, is the act and discipline Discipline is action ACTION is a bus operator in Canberra Canberra ( ) is the capital city of Austra ...
* Euler
boundary representation 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, which ...
* List of CAx companies * PLaSM – Programming Language of Solid Modeling. *
Technical drawing Technical drawing, drafting or drawing, is the act and discipline Discipline is action ACTION is a bus operator in Canberra Canberra ( ) is the capital city of Australia. Founded following the Federation of Australia, federat ...

# References

sgCore C++/C# library

The Solid Modeling Association
{{DEFAULTSORT:Solid Modeling 3D computer graphics Computer-aided design Euclidean solid geometry mk:Solid Modeling