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Geometric modeling is a branch of
applied mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
and
computational geometry that studies methods and
algorithms
In mathematics and computer science, an algorithm () is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for per ...
for the mathematical description of
shapes.
The shapes studied in geometric modeling are mostly two- or three-
dimension
In physics and mathematics, the dimension of a mathematical 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 (1D) because only one coo ...
al (''
solid figure
Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space).
A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its inte ...
s''), 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
Typography is the art and technique of Typesetting, arranging type to make written language legibility, legible, readability, readable and beauty, appealing when displayed. The arrangement of type involves selecting typefaces, Point (typogra ...
and
technical drawing
Technical drawing, drafting or drawing, is the act and discipline of composing drawings that visually communicate how something functions or is constructed.
Technical drawing is essential for communicating ideas in industry and engineering. ...
.
Three-dimensional models are central to
computer-aided design
Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve c ...
and
manufacturing
Manufacturing is the creation or production of goods with the help of equipment, labor, machines, tools, and chemical or biological processing or formulation. It is the essence of the
secondary sector of the economy. The term may refer ...
(CAD/CAM), and widely used in many applied technical fields such as
civil and
mechanical engineering
Mechanical engineering is the study of physical machines and mechanism (engineering), mechanisms that may involve force and movement. It is an engineering branch that combines engineering physics and engineering mathematics, mathematics principl ...
,
architecture
Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and construction, constructi ...
,
geology
Geology (). is a branch of natural science concerned with the Earth and other astronomical objects, the rocks of which they are composed, and the processes by which they change over time. Modern geology significantly overlaps all other Earth ...
and
medical image processing.
Geometric models are usually distinguished from
procedural and
object-oriented model
Object-oriented programming (OOP) is a programming paradigm based on the concept of ''Object (computer science), objects''. Objects can contain data (called Field (computer science), fields, Attribute (computing), attributes or Property (progra ...
s, which define the shape implicitly by an opaque
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
that generates its appearance. They are also contrasted with
digital image
A digital image is an image composed of picture elements, also known as pixels, each with '' finite'', '' discrete quantities'' of numeric representation for its intensity or gray level that is an output from its two-dimensional functions f ...
s and
volumetric models which represent the shape as a subset of a fine regular partition of space; and with
fractal
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
models that give an infinitely recursive definition of the shape. However, these distinctions are often blurred: for instance, a
digital image
A digital image is an image composed of picture elements, also known as pixels, each with '' finite'', '' discrete quantities'' of numeric representation for its intensity or gray level that is an output from its two-dimensional functions f ...
can be interpreted as a collection of
color
Color (or colour in English in the Commonwealth of Nations, Commonwealth English; American and British English spelling differences#-our, -or, see spelling differences) is the visual perception based on the electromagnetic spectrum. Though co ...
ed
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
s; and geometric shapes such as
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
s are defined by implicit mathematical equations. Also, a
fractal
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
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 and the Bézier award.
See also
*
2D geometric modeling
*
Architectural geometry
*
Computational conformal geometry
*
Computational topology
*
Computer-aided engineering
*
Computer-aided manufacturing
Computer-aided manufacturing (CAM) also known as computer-aided modeling or computer-aided machining is the use of software to control machine tools in the manufacturing of work pieces. This is not the only definition for CAM, but it is the most ...
*
Digital geometry
Digital geometry deals with discrete sets (usually discrete point 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. ...
*
Geometric modeling kernel
A geometric modeling kernel is a solid modeling software component used in computer-aided design (CAD) packages. Available modelling kernels include:
*ACIS is developed and licensed by Spatial Corporation of Dassault Systèmes.
*SMLib is develope ...
*
List of interactive geometry software
Interactive geometry software (IGS) or dynamic geometry environments (DGEs) are computer programs which allow one to create and then manipulate Geometry, geometric constructions, primarily in Euclidean plane geometry, plane geometry. In most IGS, o ...
*
Parametric equation
In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters.
In the case ...
*
Parametric surface
*
Solid modeling
*
Space partitioning
References
Further reading
General textbooks:
* This book is out of print and freely available from the author.
*
*
*
*
For multi-resolution (multiple
level of detail) geometric modeling :
*
*
Subdivision methods (such as
subdivision surface
In the field of 3D computer graphics, a subdivision surface (commonly shortened to SubD surface or Subsurf) is a curved Computer representation of surfaces, surface represented by the specification of a coarser polygon mesh and produced by a re ...
s):
*
*
*
External links
Geometry and Algorithms for CAD (Lecture Note, TU Darmstadt)
Geometric algorithms
Computer-aided design
Applied geometry
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