Solid modeling: Difference between revisions

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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 ''[[Region (geometry)|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 2-d set by taking the closure of its interior]]