Geometric design
Encyclopedia
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...
. It deals with the construction and representation of free-form curves, surfaces, or volumes. Core problems are curve and surface modelling and representation. GD studies especially the construction and manipulation of curves and surfaces given by a set of points using polynomial, rational, piecewise polynomial, or piecewise rational methods. The most important instruments here are parametric curves and parametric surface
Parametric surface
A 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, such as Bezier curve
Bézier curve
A Bézier curve is a parametric curve frequently used in computer graphics and related fields. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case....
s, spline
Spline (mathematics)
In mathematics, a spline is a sufficiently smooth piecewise-polynomial function. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low-degree polynomials, while avoiding Runge's phenomenon for higher...
curves and surfaces. An important non-parametric approach is the level set method
Level set method
The level set method is a numerical technique for tracking interfaces and shapes. The advantage of the level set method is that one can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects...
.
Application areas include shipbuilding
Shipbuilding
Shipbuilding is the construction of ships and floating vessels. It normally takes place in a specialized facility known as a shipyard. Shipbuilders, also called shipwrights, follow a specialized occupation that traces its roots to before recorded history.Shipbuilding and ship repairs, both...
, aircraft
Aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air, or, in general, the atmosphere of a planet. An aircraft counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines.Although...
, and automotive industries
Automotive industry
The automotive industry designs, develops, manufactures, markets, and sells motor vehicles, and is one of the world's most important economic sectors by revenue....
, as well as architectural design
Architectural Design
Architectural Design, also known as AD, is a UK-based architectural journal first launched in 1930.In its early days it was more concerned with the British scene, but gradually became more international. It also moved away from presenting mostly news towards theme-based issues...
. The modern ubiquity and power of computers means that even perfume bottles and shampoo dispensers are designed using techniques unheard of by shipbuilders of 1960s.
Geometric models can be built for objects of any 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...
in any geometric space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
. Both 2D
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...
and 3D geometric models
3D modeling
In 3D computer graphics, 3D modeling is the process of developing a mathematical representation of any three-dimensional surface of object via specialized software. The product is called a 3D model...
are extensively used in computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
. 2D 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....
. 3D models
3D modeling
In 3D computer graphics, 3D modeling is the process of developing a mathematical representation of any three-dimensional surface of object via specialized software. The product is called a 3D model...
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...
, and many applied technical fields such as geology and medical image processing.
Geometric models are usually distinguished from procedural
Procedural modeling
Procedural modeling is an umbrella term for a number of techniques in computer graphics to create 3D models and textures from sets of rules. L-Systems, fractals, and generative modeling are procedural modeling techniques since they apply algorithms for producing scenes...
and object-oriented models
Object-Oriented Modeling
Object-oriented modeling , also called object-oriented programming is a modeling paradigm mainly used in computer programming. Prior to the rise of OOM, the dominant paradigm was procedural programming, which emphasized the use of discrete reusable code blocks that could stand on their own, take...
, which define the shape implicitly by an 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...
. 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; and with implicit mathematical models such as the zero set of an arbitrary polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
. However, the distinction is often blurred: for instance, geometric shapes can be represented by objects; 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, the modeling of 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...
objects often requires a combination of geometric and procedural techniques.
Geometric problems originating in architecture can lead to interesting research and results in geometry processing, computer-aided geometric design, and discrete differential geometry.
See also
- List of interactive geometry software
- CAD/CAMComputer-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...
/CAEComputer-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...