Computer-aided geometric design
Computer-aided geometric design (CAGD), also known as geometric modelling, is a branch of computational geometry. It deals with the construction and representation of free-form curves, surfaces, or volumes.[1] Core problems are curve and surface modelling and representation.
The most important instruments here are parametric curves and parametric surfaces, such as Bezier curves, spline curves and surfaces. An important non-parametric approach is the level set method.
Application areas include shipbuilding, aircraft, and automotive industries. 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 any geometric space. Both 2D and 3D geometric models are extensively used in computer graphics. 2D models are important in computer typography and technical drawing. 3D models are central to computer-aided design and manufacturing, and many applied technical fields such as geology and medical image processing.
Geometric models are usually distinguished from procedural and object-oriented models, which define the shape implicitly by an algorithm. They are also contrasted with digital images and volumetric models; and with implicit mathematical models such as the zero set of an arbitrary polynomial. However, the distinction is often blurred: for instance, geometric shapes can be represented by objects; a digital image 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 objects often requires a combination of geometric and procedural techniques.
Computer-aided geometric design (CAGD) 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. This branch is closely related to several other branches, such as geometric modeling (for example, Non-Uniform Rational B-Spline (NURBS) objects represent the fundamental structures of modern computer systems used in the aircraft and car industry, such as CATIA V5) or data fitting (interpolation, approximation of a set of points).[2]
Geometric problems originating in architecture can lead to interesting research and results in geometry processing, computer-aided geometric design, and discrete differential geometry.[3]
References
- ^ Farin, G.: A History of Curves and Surfaces in CAGD, Handbook of Computer Aided Geometric Design
- ^ Computer Aided Geometric Design in Mathematica
- ^ H. Pottmann, S. Brell-Cokcan and J. Wallner:Discrete surfaces for architectural design
Journals
See also
- CAD/CAM/CAE
- Solid modeling
- Computational topology
- Digital geometry
- Computational Geometry Algorithms Library (CGAL)
- Space partitioning
- Wikiversity:Topic:Computational geometry
- parametric curves
- parametric surfaces
- Architectural geometry
External links
- Geometric Modeling and Industrial Geometry
- K3DSurf — A program to visualize and manipulate Mathematical models in three, four, five and six dimensions. K3DSurf supports Parametric equations and Isosurfaces
- JavaView — a 3D geometry viewer and a mathematical visualization software.
- Related Wolfram Demonstration Projects