Editing Solid modeling (section)

== 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.<ref name = "First Principles">{{cite journal |title= Solid Modeling: Current Status and Research Directions|journal = IEEE Computer Graphics and Applications|volume = 3|issue = 7|pages = 25–37|author1=Requicha, A.A.G  |author2=Voelcker, H.  |name-list-style=amp |year= 1983 |publisher= IEEE Computer Graphics |doi= 10.1109/MCG.1983.263271|s2cid = 14462567}}</ref>

All manufactured components have finite size and well behaved [[Boundary (topology)|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.

[[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'' ⊂ <math>\mathbb{R}^3</math> 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 <math>\mathbb{R}^3</math> 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'' ⊂ <math>\mathbb{R}^3</math> 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 <math>\mathbb{R}^3</math> (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 ∪<sup>∗</sup>, ∩<sup>∗</sup>, and −<sup>∗</sup>.

The combinatorial characterization of a set ''X'' ⊂ <math>\mathbb{R}^3</math> 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=http://pi.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 <math>\mathbb{R}^n</math> 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.