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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.
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#REDIRECT [[Computer-aided geometric design]]

The shapes studied in geometric modeling are mostly two- or three-[[dimension]]al, although many of its tools and principles can be applied to sets of any finite dimension. [[2D geometric model|Two-dimensional model]]s are important in computer [[typography]] and [[technical drawing]]. [[3D geometric model|Three-dimensional model]]s are central to [[computer-aided geometric design]] and [[computer-aided manufacturing|manufacturing]], and many applied technical fields such as [[civil engineering|civil]] and [[mechanical engineering]], [[architecture]], [[geologic modeling|geology]] and [[medical image processing]]. <ref>Farin, G.: A History of Curves and Surfaces in CAGD, [http://books.google.com/books?id=0SV5G8fgxLoC&printsec=frontcover&dq=Computer+Aided+GEOMETRIC+DESIGN&source=gbs_summary_s&cad=0 Handbook of Computer Aided Geometric Design]</ref>

Most geometric models are developed and used with computers, and therefore with techniques of [[computational geometry]] and [[numerical methods]].

Geometric models are usually distinguished from [[procedural model|procedural]] and [[object-oriented model]]s, which define the shape implicitly by an [[algorithm]] that. 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. 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, the modeling of [[fractal]] objects often requires a combination of geometric and procedural techniques.

'''Computer-aided geometric design''' ('''CAGD''') is the technology of representing, building and manipulating [[geometric model]]s of shapes curves, surfaces, or volumes using computers.

==Types of geometric models==

In a [[parametric model (geometry)|parametric model]], the desired shape is defined by a [[function (mathematics)|function]] ''F'' from some simple domain — such as a [[square (geometry)|square]] or [[triangle]] — into two- or three-dimensional space. Specifically, the shape is the set of all ponts ''F''(''p'') where ''p'' ranges over the domain of ''F''.

In an [[implicit model (geometry)|implicit model]], on the other hand, the shape is defined by an equation ''F''(''p'') = 0 or F''(''p'') &lt; 0, for some function ''F'' from two- or three-dimensional space to the real numbers. This technique is also called the [[zero set]] or [[level set method]].

In both cases, the function ''F'' has a simple mathematical expression, such as a polynomial, a rational function, or a trigonometric expression. These simple moedls allow limited control of the shape, by adjusting the coefficients of the formula. In many applications, such computer-aided design, many of thse simple models are combined into a ''free-form'' model, which allow direct and arbirarily detailed control of the shape. These are often represented by piecewise [[parametric curve]]s or [[parametric surface|surfaces]], with [[polynomial]] or [[rational function|rational parts]], such as [[Bezier curve]]s, [[spline (mathematics)|spline]] curves and surfaces.<ref>H. Pottmann, S. Brell-Cokcan, and J. Wallner:[http://www.geometrie.tuwien.ac.at/ig/sn/2007/pbw_surfaces_07/pbw_surfaces_07.html ''Discrete surfaces for architectural design''] </ref>

==References==
{{Reflist}}

==See also==
* [[Computer-aided manufacturing]]
* [[Computer-aided engineering]]
* [[Solid modeling]]
* [[Computational topology]]
* [[Digital geometry]]
* [[CGAL|Computational Geometry Algorithms Library (CGAL)]]
* [[Space partitioning]]
* [[Wikiversity:Topic:Computational geometry]]
* [[Parametric curve]]s
* [[Parametric surface]]s
* [[Architectural geometry]]

==External links==

*[http://www.geometrie.tuwien.ac.at/ig/ Geometric Modeling and Industrial Geometry ]
*[http://k3dsurf.sourceforge.net/ K3DSurf &mdash; A program to visualize and manipulate Mathematical models in three, four, five and six dimensions. K3DSurf supports Parametric equations and Isosurfaces]
*[http://www.javaview.de/ JavaView &mdash; a 3D geometry viewer and a mathematical visualization software.]
*[http://demonstrations.wolfram.com/topic.html?topic=3D+Graphics&limit=100 Related Wolfram Demonstration Projects]

[[Category:Geometric algorithms]]
[[Category:Computational science]]
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[[Category:Computer-aided design]]

[[de:Geometrische Modellierung]]
[[no:Geometrisk modellering]]
[[zh:几何模型]]

Revision as of 22:36, 24 April 2009

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-dimensional, although many of its tools and principles can be applied to sets of any finite dimension. Two-dimensional models are important in computer typography and technical drawing. Three-dimensional models are central to computer-aided geometric design and manufacturing, and many applied technical fields such as civil and mechanical engineering, architecture, geology and medical image processing. [1]

Most geometric models are developed and used with computers, and therefore with techniques of computational geometry and numerical methods.

Geometric models are usually distinguished from procedural and object-oriented models, which define the shape implicitly by an algorithm that. They are also contrasted with digital images and volumetric models which represent the shape as a subset of a fine regular partition of space. However, these distinctions are often blurred: for instance, a digital image can be interpreted as a collection of colored squares; and geometric shapes such as circles are defined by implicit mathematical equations. Also, the modeling of fractal objects often requires a combination of geometric and procedural techniques.

Computer-aided geometric design (CAGD) is the technology of representing, building and manipulating geometric models of shapes curves, surfaces, or volumes using computers.

Types of geometric models

In a parametric model, the desired shape is defined by a function F from some simple domain — such as a square or triangle — into two- or three-dimensional space. Specifically, the shape is the set of all ponts F(p) where p ranges over the domain of F.

In an implicit model, on the other hand, the shape is defined by an equation F(p) = 0 or F(p) < 0, for some function F from two- or three-dimensional space to the real numbers. This technique is also called the zero set or level set method.

In both cases, the function F has a simple mathematical expression, such as a polynomial, a rational function, or a trigonometric expression. These simple moedls allow limited control of the shape, by adjusting the coefficients of the formula. In many applications, such computer-aided design, many of thse simple models are combined into a free-form model, which allow direct and arbirarily detailed control of the shape. These are often represented by piecewise parametric curves or surfaces, with polynomial or rational parts, such as Bezier curves, spline curves and surfaces.[2]

References

  1. ^ Farin, G.: A History of Curves and Surfaces in CAGD, Handbook of Computer Aided Geometric Design
  2. ^ H. Pottmann, S. Brell-Cokcan, and J. Wallner:Discrete surfaces for architectural design

See also

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