Geometric modeling

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. 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 Bezier award.^[3]

See also

• Geometric modeling kernel
• Architectural geometry
• Сomputational conformal geometry
• Computer-aided engineering
• Computer-aided manufacturing
• Computational topology
• Digital geometry
• List of interactive geometry software
• Parametric curves
• Parametric surfaces
• Pythagorean-hodograph curves
• Solid modeling
• Space partitioning

References

Further reading

General textbooks:

• Jean Gallier (1999). Curves and Surfaces in Geometric Modeling: Theory and Algorithms. Morgan Kaufmann.  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)