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 paramettric or implicit model when its recursive definition is truncated to a finite depth.
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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
- ^ Farin, G.: A History of Curves and Surfaces in CAGD, Handbook of Computer Aided Geometric Design
- ^ H. Pottmann, S. Brell-Cokcan, and J. Wallner:Discrete surfaces for architectural design
See also
- Computer-aided manufacturing
- Computer-aided engineering
- Solid modeling
- Computational topology
- Digital geometry
- Computational Geometry Algorithms Library (CGAL)
- Space partitioning
- Wikiversity:Topic:Computational geometry
- Parametric curves
- Parametric surfaces
- Architectural geometry
External links
- Geometric Modeling and Industrial Geometry
- K3DSurf — A program to visualize and manipulate Mathematical models in three, four, five and six dimensions. K3DSurf supports Parametric equations and Isosurfaces
- JavaView — a 3D geometry viewer and a mathematical visualization software.
- Related Wolfram Demonstration Projects
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