Geometric modeling: Difference between revisions

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

'''Geometric modeling''' is a branch of [[applied mathematics]] and [[computational geometry]] that studies methods and [[algorithms]] for the mathematical description of [[Shape (mathematics)|shapes]].

The shapes studied in geometric modeling are mostly two- or three-[[dimension]]al (''[[solid figure]]s''), although many of its tools and principles can be applied to sets of any finite dimension. Today most geometric modeling is done with computers and for computer-based applications. [[2D geometric model|Two-dimensional model]]s are important in computer [[typography]] and [[technical drawing]]. [[3D modeling|Three-dimensional model]]s are central to [[computer-aided design]] and [[computer-aided manufacturing|manufacturing]] (CAD/CAM), and widely used in many applied technical fields such as [[civil engineering|civil]] and [[mechanical engineering]], [[architecture]], [[geologicgeological modelingmodelling|geology]] and [[medical image processing]].<ref>Handbook of Computer Aided Geometric Design</ref>

Geometric models are usually distinguished from [[procedural modelmodeling|procedural]] and [[object-oriented modeling|object-oriented model]]s, which define the shape implicitly by an opaque [[algorithm]] that generates its appearance.{{cncitation needed|date=August 2014}} They are also contrasted with [[digital image]]s and [[volumetric model]]s which represent the shape as a subset of a fine regular partition of space; and with [[fractal]] models that give an infinitely recursive definition of the shape. However, these distinctions are often blurred: for instance, a [[digital image]] can be interpreted as a collection of [[color]]ed [[square (geometry)|square]]s; and geometric shapes such as [[circle]]s are defined by implicit mathematical equations. Also, a [[fractal]] model yields a parametric or implicit model when its recursive definition is truncated to a finite depth.

Notable awards of the area are the John A. Gregory Memorial Award<ref>{{Cite web |url=http://geometric-modelling.org |access-date=2025-07-08 |website=geometric-modelling.org | title=John A. Gregory Memorial Award}}</ref> and the BezierBézier award.<ref>{{Cite web |url=http://www.solidmodeling.org/bezier_award.html |title=Archived copy |access-date=2014-06-20 |archive-date=2014-07-15 |archive-url=https://web.archive.org/web/20140715121544/http://www.solidmodeling.org/bezier_award.html |url-status=dead }}</ref>

==See also==

* [[2D geometric modeling]]

* [[Architectural geometry]]

* computational [[Conformal geometry|Computational conformal geometry]]

* [[Computational topology]]

* [[Computer-aided engineering]]

* [[Computer-aided manufacturing]]

* [[Computational topology]]

* [[Digital geometry]]

* [[Geometric modeling kernel]]

* [[List of interactive geometry software]]

* [[Parametric curveequation]]s

* [[Parametric surface]]s

* [[Pythagorean-hodograph curve]]s

* [[Solid modeling]]

* [[Space partitioning]]

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==Further reading==

General textbooks:

* {{cite book|url=http://www.cis.upenn.edu/~jean/gbooks/geom1.html|title=Curves and Surfaces in Geometric Modeling: Theory and Algorithms|author=Jean Gallier|authorlink= Jean Gallier |publisher=Morgan Kaufmann|year=1999}} This book is out of print and freely available from the author.

* {{cite book|author=Gerald E. Farin|title=Curves and Surfaces for CAGD: A Practical Guide|year=2002|publisher=Morgan Kaufmann|isbn=978-1-55860-737-8|edition=5th|url=http://www.farinhansford.com/books/cagd/}}

* {{cite book|author=Max K. Agoston|title=Computer Graphics and Geometric Modelling: Mathematics|year=2005|publisher=Springer Science & Business Media|isbn=978-1-85233-817-6}} and its companion {{cite book|author=Max K. Agoston|title=Computer Graphics and Geometric Modelling: Implementation & Algorithms|year=2005|publisher=Springer Science & Business Media|isbn=978-1-84628-108-2}}

* {{cite book|author=Michael E. Mortenson|title=Geometric Modeling|year=2006|publisher=Industrial Press|isbn=978-0-8311-3298-9|edition=3rd}}

* {{cite book|author=Ronald Goldman|authorlink=Ron Goldman (mathematician)|title=An Integrated Introduction to Computer Graphics and Geometric Modeling|year=2009|publisher=CRC Press|isbn=978-1-4398-0334-9|edition=1st}}

* {{cite book|author=Nikolay N. Golovanov |title=Geometric Modeling: The mathematics of shapes |url=http://www.amazon.com/Geometric-Modeling-The-mathematics-shapes/dp/1497473195 |publisher=[[CreateSpace Independent Publishing Platform]] |isbn=978-1497473195 |year=2014}}

For multi-resolution (multiple [[Level of detail (computer graphics)|level of detail]]) geometric modeling :

* {{cite book|author1=Armin Iske|author2=Ewald Quak|author3=Michael S. Floater|title=Tutorials on Multiresolution in Geometric Modelling: Summer School Lecture Notes|year=2002|publisher=Springer Science & Business Media|isbn=978-3-540-43639-3}}

* {{cite book|author1=Neil Dodgson|author2=Michael S. Floater|author3=Malcolm Sabin|title=Advances in Multiresolution for Geometric Modelling|year=2006|publisher=Springer Science & Business Media|isbn=978-3-540-26808-6}}

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==External links==

* [http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf Geometry and Algorithms for CAD ] (Lecture Note, TU Darmstadt)

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[[Category:Geometric algorithms]]

[[Category:Computer-aided design]]

[[Category:Applied geometry]]