# 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

- ↑ Handbook of Computer Aided Geometric Design
- ↑ http://geometric-modelling.org
- ↑ http://www.solidmodeling.org/bezier_award.html

## 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)