Geometric modeling
Encyclopedia
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-dimension
al, 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 model
s 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.
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 image
s 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 color
ed square
s; and geometric shapes such as circle
s 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.
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
and computational geometry
Computational geometry
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational...
that studies methods and algorithms for the mathematical description of shapes.
The shapes studied in geometric modeling are mostly two- or three-dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al, 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 model
2D geometric model
A 2D geometric model is a geometric model of an object as two-dimensional figure, usually on the Euclidean or Cartesian plane.Even though all material objects are three-dimensional, a 2D geometric model is often adequate for certain flat objects, such as paper cut-outs and machine parts made of...
s are important in computer typography
Typography
Typography is the art and technique of arranging type in order to make language visible. The arrangement of type involves the selection of typefaces, point size, line length, leading , adjusting the spaces between groups of letters and adjusting the space between pairs of letters...
and technical drawing
Technical drawing
Technical drawing, also known as drafting or draughting, is the act and discipline of composing plans that visually communicate how something functions or has to be constructed.Drafting is the language of industry....
. Three-dimensional models are central to computer-aided design
Computer-aided design
Computer-aided design , also known as computer-aided design and drafting , is the use of computer technology for the process of design and design-documentation. Computer Aided Drafting describes the process of drafting with a computer...
and manufacturing
Computer-aided manufacturing
Computer-aided manufacturing is the use of computer software to control machine tools and related machinery in the manufacturing of workpieces. This is not the only definition for CAM, but it is the most common; CAM may also refer to the use of a computer to assist in all operations of a...
(CAD/CAM), and widely used in many applied technical fields such as civil
Civil engineering
Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including works like roads, bridges, canals, dams, and buildings...
and mechanical engineering
Mechanical engineering
Mechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the production and usage of heat and mechanical power for the...
, architecture
Architecture
Architecture is both the process and product of planning, designing and construction. Architectural works, in the material form of buildings, are often perceived as cultural and political symbols and as works of art...
, geology and medical image processing.
Geometric models are usually distinguished from procedural and object-oriented models, which define the shape implicitly by an opaque algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
that generates its appearance. They are also contrasted with digital image
Digital image
A digital image is a numeric representation of a two-dimensional image. Depending on whether or not the image resolution is fixed, it may be of vector or raster type...
s and volumetric models which represent the shape as a subset of a fine regular partition of space; and with fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
models that give an infinitely recursive definition of the shape. However, these distinctions are often blurred: for instance, a digital image
Digital image
A digital image is a numeric representation of a two-dimensional image. Depending on whether or not the image resolution is fixed, it may be of vector or raster type...
can be interpreted as a collection of color
Color
Color or colour is the visual perceptual property corresponding in humans to the categories called red, green, blue and others. Color derives from the spectrum of light interacting in the eye with the spectral sensitivities of the light receptors...
ed square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
s; and geometric shapes such as circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
s are defined by implicit mathematical equations. Also, a fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
model yields a parametric or implicit model when its recursive definition is truncated to a finite depth.
See also
- List of interactive geometry software
- Computer-aided manufacturingComputer-aided manufacturingComputer-aided manufacturing is the use of computer software to control machine tools and related machinery in the manufacturing of workpieces. This is not the only definition for CAM, but it is the most common; CAM may also refer to the use of a computer to assist in all operations of a...
- Computer-aided engineeringComputer-aided engineeringComputer-aided engineering is the broad usage of computer software to aid in engineering tasks. It includes computer-aided design , computer-aided analysis , computer-integrated manufacturing , computer-aided manufacturing , material requirements planning , and computer-aided planning .- Overview...
- Solid modelingSolid modelingSolid modeling is a consistent set of principles for mathematical and computer modeling of three dimensional solids. Solid modeling is distinguished from related areas of Geometric modeling and Computer graphics by its emphasis on physical fidelity...
- Computational topology
- Digital geometryDigital geometryDigital geometry deals with discrete sets considered to be digitized models or images of objects of the 2D or 3D Euclidean space.Simply put, digitizing is replacing an object by a discrete set of its points...
- Space partitioningSpace partitioningIn mathematics, space partitioning is the process of dividing a space into two or more disjoint subsets . In other words, space partitioning divides a space into non-overlapping regions...
- Wikiversity:Topic:Computational geometry
- Parametric curves
- Parametric surfaceParametric surfaceA parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters. Parametric representation is the most general way to specify a surface. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence...
s - Architectural geometryArchitectural geometryArchitectural geometry is an area of research which combines applied geometry and architecture, which looks at the design, analysis and manufacture processes...
External links
- Geometric Modeling and Industrial Geometry
- Related Wolfram Demonstration Projects
- K. T. Wong, Y. I. Wu & M. Abdulla, “Landmobile Radiowave Multipaths' DOA-Distribution: Assessing Geometric Models by the Open Literature's Empirical Datasets,” IEEE Transactions on Antennas & Propagation, vol. 58, no. 2, pp. 946-958, February 2010.http://www.eie.polyu.edu.hk/~enktwong/ktw/WongKT_APT0310.pdf