Solid modeling: Difference between revisions

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{{Short description|Set of principles for modeling solid geometry}}

{{Technical|date=January 2021}}

{{MOS|article|[[MOS:BBB]]|date=May 2024}}

{{Use dmy dates|date=February 2020}}

[[File:Jack-in-cube solid model, light background.gif|thumb|right|The geometry in solid modeling is fully described in 3{{nbhyph}}D space; objects can be viewed from any angle.]]

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[[File:Regularize1.png|thumb|right|450px|Regularization of a 2D set by taking the closure of its interior]]

According to the continuum point-set model of solidity, all the points of any ''X'' ⊂ ℝ<supmath>\mathbb{R}^3</supmath> can be classified according to their ''[[Neighborhood (topology)|neighborhoods]]'' with respect to ''X'' as ''[[Interior (topology)|interior]]'', ''[[Exterior (topology)|exterior]]'', or ''[[Boundary (topology)|boundary]]'' points. Assuming ℝ<supmath>\mathbb{R}^3</supmath> is endowed with the typical [[Euclidean metric]], a neighborhood of a point ''p'' ∈''X'' takes the form of an [[Ball (mathematics)|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 (topology)|closure]]'' of their interior. Any ''X'' ⊂ ℝ<supmath>\mathbb{R}^3</supmath> 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<supmath>\mathbb{R}^3</supmath> (by the [[Heine–Borel theorem|Heine-Borel theorem]] it is implied that all solids are [[Compact space|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.<ref name = "Regularized operations">{{citation|doi=10.1016/0010-4485(80)90025-1|title=Closure of Boolean operations on geometric entities|journal=Computer-Aided Design|volume=12|issue=5|pages=219–220|year=1980|last1=Tilove|first1=R.B.|last2=Requicha|first2=A.A.G.}}</ref> The regularized set operations are denoted ∪^∗, ∩^∗, and −^∗.

The combinatorial characterization of a set ''X'' ⊂ ℝ<supmath>\mathbb{R}^3</supmath> as a solid involves representing ''X'' as an orientable [[CW complex|cell complex]] so that the cells provide finite spatial addresses for points in an otherwise innumerable continuum.<ref name="Solid Modeling"/> The class of [[semi-analytic]] [[Bounded set|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 [[Stratification (mathematics)|stratified]] into a collection of disjoint cells of dimensions 0,1,2,3. A [[Triangulation (topology)|triangulation]] of a semi-analytic set into a collection of points, [[line segment]]s, triangular [[face (geometry)|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.<ref name = "Representations">{{cite journal |title= Representations for Rigid Solids: Theory, Methods, and Systems|journal= ACM Computing Surveys|volume= 12|issue= 4|pages= 437–464|author= Requicha, A.A.G. |year= 1980 |doi= 10.1145/356827.356833|s2cid= 207568300}}</ref> In particular this implies the [[Euler characteristic]] of the combinatorial boundary<ref name = "Hatcher">{{cite book |url=httphttps://wwwpi.math.cornell.edu/~hatcher/AT/ATpage.html |title= Algebraic Topology|author= Hatcher, A. |year= 2002 |publisher= Cambridge University Press |access-date=20 April 2010}}</ref> 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 curve theorem|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 ℝ<supmath>''\mathbb{R}^n''</supmath> 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]]. All representation schemes are organized in terms of a finite number of operations on a set of primitives. TTherefore, 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 [[Constructive solid geometry|combinations of regularized booleanBoolean operations]] cannot necessarily be represented as the [[Solid sweep|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 techniques used to create or represent solid models.<ref name="Representations"/> Modern modeling software may use a combination of these schemes to represent a solid.

=== Primitive instancing===

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===Implicit representation===

{{Main|Function representation}}

A very general method of defining a set of points ''X'' is to specify a [[Predicate (mathematical logic)|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 <math>f= ax + by + cz + d</math> the conditions <math>f(p) =0</math>, <math> f(p) > 0</math>, and <math>f(p) < 0</math> represent, respectively, a plane and two open linear [[Half-space (geometry)|halfspaces]]. More complex functional primitives may be defined by booleanBoolean combinations of simpler predicates. Furthermore, the theory of [[Rvachev function|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.

===Parametric and feature-based modeling===

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== History of solid modelers ==

{{Unreferenced section|date=January 2012}}

{{AdvertPromotional|section|date=June 2015}}

The historical development of solid modelers has to be seen in context of the whole [[Computer-aided design|history of CAD]], the key milestones being the development of the research system BUILD followed by its commercial spin-off [[Romulus (b-rep solid modeler)|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.<ref>{{cite journal

|last = Yares

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{{Main|Computer-aided design}}

{{More citations needed section|date=January 2012}}

The modeling of solids is only the minimum requirement of a [[Computer-aided design#Capabilities|CAD system's capabilities]]. Solid modelers have become commonplace in engineering departments in the last ten years{{When|date=December 2011}} due to faster computers and competitive software pricing. Solid modeling software creates a virtual 3D representation of components for machine design and analysis.<ref name="LaCourse Handbook">{{cite book|last=LaCourse|first=Donald|title=Handbook of Solid Modeling|publisher=McGraw Hill|year=1995|pages=2.5|chapter=2|isbn=978-0-07-035788-4}}</ref> A typical [[GUI|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 modelling|assembly modeling]] methods. An assembly model incorporates references to individual part models that comprise the product.<ref name="LaCourse Handbook 11.3">{{cite book|last=LaCourse|first=Donald|title=Handbook of Solid Modeling|publisher=McGraw Hill|year=1995|pages=111.2|chapter=11|isbn=978-0-07-035788-4}}</ref>

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* [[Engineering drawing]]

* Euler [[boundary representation]]

* [[List of CAx companies]]

* [[PLaSM]] – Programming Language of Solid Modeling.

* [[Technical drawing]]