Geometrical design (GD) is a branch of

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 ar ...

. It deals with the construction and representation of free-form curves, surfaces, or volumes and is closely related to

geometric modeling __NOTOC__ 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-dimensio ...

. 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 obj ...

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 occ ...

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 o ...

. 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 coord ...

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 de ...

. 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 discipline of composing drawings that visually communicate how something functions or is constructed. Technical drawing is essential for communicating ideas in industry and engineering ...

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, an ...

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 co ...

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 ...

, and many applied technical fields such as

geology Geology () is a branch of natural science concerned with Earth and other Astronomical object, astronomical objects, the features or rock (geology), rocks of which it is composed, and the processes by which they change over time. Modern geology ...

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 rev ...

. Geometric models are usually distinguished from procedural 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 ...

s and

volumetric model In 3D computer graphics and 3D modeling, 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 in ...

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 e ...

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 exampl ...

. However, the distinction is often blurred: for instance, geometric shapes can be represented by

objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...

; a

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 ill ...

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 ch ...

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 ...

References

External links

Evolute Research and ConsultingComputer Aided Geometric Design
{{Design Geometric algorithms Computational science Computer-aided design Applied geometry

See also

References

External links