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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.

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. Today most geometric modeling is done with computers and for computer-based applications. Two-dimensional models are important in computer typography and technical drawing. Three-dimensional models are central to computer-aided design and manufacturing (CAD/CAM), and widely used in many applied technical fields such as civil and mechanical engineering, architecture, geology and medical image processing.^[1]

Geometric models are usually distinguished from procedural and object-oriented models, which define the shape implicitly by an opaque algorithm that generates its appearance.^{[citation needed]} They are also contrasted with digital images and volumetric models 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 colored squares; and geometric shapes such as circles 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^[2] and the Bézier award.^[3]

See also

• Architectural geometry


• Computational conformal geometry


• Computational topology


• Computer-aided engineering


• Computer-aided manufacturing


• Digital geometry


• Geometric modeling kernel


• List of interactive geometry software


• Parametric equation


• Parametric surface


• Solid modeling


• Space partitioning

References

Further reading

General textbooks:

• 
  Jean Gallier (1999). Curves and Surfaces in Geometric Modeling: Theory and Algorithms. Morgan Kaufmann..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em} This book is out of print and freely available from the author.


• 
  Gerald E. Farin (2002). Curves and Surfaces for CAGD: A Practical Guide (5th ed.). Morgan Kaufmann. ISBN 978-1-55860-737-8.
  


• 
  Max K. Agoston (2005). Computer Graphics and Geometric Modelling: Mathematics. Springer Science & Business Media. ISBN 978-1-85233-817-6. and its companion Max K. Agoston (2005). Computer Graphics and Geometric Modelling: Implementation & Algorithms. Springer Science & Business Media. ISBN 978-1-84628-108-2.
  


• 
  Michael E. Mortenson (2006). Geometric Modeling (3rd ed.). Industrial Press. ISBN 978-0-8311-3298-9.
  


• 
  Ronald Goldman (2009). An Integrated Introduction to Computer Graphics and Geometric Modeling (1st ed.). CRC Press. ISBN 978-1-4398-0334-9.
  


• 
  Nikolay N. Golovanov (2014). Geometric Modeling: The mathematics of shapes. CreateSpace Independent Publishing Platform. ISBN 978-1497473195.
  

For multi-resolution (multiple level of detail) geometric modeling :

• 
  Armin Iske; Ewald Quak; Michael S. Floater (2002). Tutorials on Multiresolution in Geometric Modelling: Summer School Lecture Notes. Springer Science & Business Media. ISBN 978-3-540-43639-3.
  


• 
  Neil Dodgson; Michael S. Floater; Malcolm Sabin (2006). Advances in Multiresolution for Geometric Modelling. Springer Science & Business Media. ISBN 978-3-540-26808-6.
  

Subdivision methods (such as subdivision surfaces):

• 
  Joseph D. Warren; Henrik Weimer (2002). Subdivision Methods for Geometric Design: A Constructive Approach. Morgan Kaufmann. ISBN 978-1-55860-446-9.
  


• 
  Jörg Peters; Ulrich Reif (2008). Subdivision Surfaces. Springer Science & Business Media. ISBN 978-3-540-76405-2.
  


• 
  Lars-Erik Andersson; Neil Frederick Stewart (2010). Introduction to the Mathematics of Subdivision Surfaces. SIAM. ISBN 978-0-89871-761-7.
  

External links

• 
  Geometry and Algorithms for CAD (Lecture Note, TU Darmstadt)

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