Geometrical design (GD) is a branch of
computational geometry. It deals with the construction and representation of free-form curves, surfaces, or volumes and is closely related to
geometric modeling
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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- dimen ...
. 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 curve
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric o ...
s and
parametric surface A parametric surface is a surface in the Euclidean space \R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that o ...
s, such as
Bézier curve
A Bézier curve ( ) is a parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real-world shape ...
s,
spline curves and surfaces. An important non-parametric approach is the
level-set method
Level-set methods (LSM) are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level-set model is that one can perform numerical computations involving curves and surfaces on a ...
.
Application areas include shipbuilding, aircraft, and automotive industries, as well as architectural design. 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
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 coor ...
in any
geometric
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
space. Both
2D and
3D geometric models are extensively used in
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great deal ...
.
2D models are important in computer
typography
Typography is the art and technique of arranging type to make written language legible, readable and appealing when displayed. The arrangement of type involves selecting typefaces, point sizes, line lengths, line-spacing ( leading), an ...
and
technical drawing
Technical drawing, drafting or drawing, is the act and Academic discipline, discipline of composing Plan (drawing), drawings that Visual communication, visually communicate how something functions or is constructed.
Technical drawing is essent ...
.
3D models
In 3D computer graphics, 3D modeling is the process of developing a mathematical coordinate-based representation of any surface of an object (inanimate or living) in three dimensions via specialized software by manipulating edges, vertices, a ...
are central to
computer-aided design 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 secondary sector of the economy. The term may refer to a ...
, and many applied technical fields such as
geology
Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ea ...
and
medical image processing
Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to revea ...
.
Geometric models are usually distinguished from
procedural
Procedural may refer to:
* Procedural generation, a term used in computer graphics applications
*Procedural knowledge, the knowledge exercised in the performance of some task
* Procedural law, a legal concept
*Procedural memory, a cognitive scienc ...
and
object-oriented models, which define the shape implicitly by an
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
. 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 model
In 3D computer graphics and modeling, volumetric meshes are a polygonal representation of the interior volume of an object. Unlike polygon meshes, which represent only the surface as polygons, volumetric meshes also discretize the interior struc ...
s; and with mathematical models such as the
zero set
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or ...
of an arbitrary
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
. However, the distinction is often blurred: for instance, geometric shapes can be represented by
objects; 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 colored squares; and geometric shapes such as circles are defined by implicit mathematical equations. Also, the modeling of
fractal
In mathematics, a fractal is a 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 scales, as il ...
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
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 discrete differential geometry.
[H. Pottmann, S. Brell-Cokcan and J. Wallner]
''Discrete surfaces for architectural design''
In architecture, geometric design is associated with the pioneering explorations of
Chuck Hoberman
Chuck Hoberman (born 1956) is an artist, engineer, architect, and inventor of folding toys and structures, most notably the Hoberman sphere.
Early life and education
Hoberman's father was an architect, and his mother, Mary Ann Hoberman, was a ...
into transformational geometry as a design idiom, and applications of this design idiom within the domain of
architectural geometry
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 constructing buildings o ...
.
See also
*
Architectural geometry
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 constructing buildings o ...
*
Computational topology Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational geometry and computational complexity theory.
A primary concern of algorithmic topology, as its ...
*
CAD
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 ...
/
CAM
Calmodulin (CaM) (an abbreviation for calcium-modulated protein) is a multifunctional intermediate calcium-binding messenger protein expressed in all eukaryotic cells. It is an intracellular target of the secondary messenger Ca2+, and the bin ...
/
CAE CAE may refer to:
Organisations Aviation
* CAE Aviation, a Luxembourgian aviation services company
* CAE Inc. (formerly Canadian Aviation Electronics), a Canadian manufacturer of simulation technologies and training provider
* Régional Compagnie A ...
*
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. The ...
*
Geometric design of roads
The geometric design of roads is the branch of highway engineering concerned with the positioning of the physical elements of the roadway according to standards and constraints. The basic objectives in geometric design are to optimize efficien ...
*
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 geometric constructions, primarily in plane geometry. In most IGS, one starts construction by putting a ...
*
Parametric curve
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric o ...
s
*
Parametric surface A parametric surface is a surface in the Euclidean space \R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that o ...
s
*
Solid modeling
Solid modeling (or solid modelling) is a consistent set of principles for mathematical and computer modeling of three-dimensional shapes '' (solids)''. Solid modeling is distinguished from related areas of geometric modeling and computer graphi ...
*
Space partitioning
In geometry, space partitioning is the process of dividing a space (usually a Euclidean space) into two or more disjoint subsets (see also partition of a set). In other words, space partitioning divides a space into non-overlapping regions. Any ...
*
Wikiversity:Topic:Computational geometry
References
External links
Evolute Research and ConsultingComputer Aided Geometric Design
{{Design
Geometric algorithms
Computational science
Computer-aided design
Applied geometry