Body (geometry)
A body is a three-dimensional figure in geometry that can be described by its surface . The surface of a body can be composed of flat or curved surface pieces. If the surface of a body consists only of flat pieces, it is a polyhedron . There are mathematical formulas for calculating the volume and surface area of many geometric bodies (see the Geometry formula collection ). More precisely, a geometric figure of the type just described is called a three-dimensional body , since this concept formation can also be generalized to higher dimensions.
definition
Geometric bodies can be mathematically defined in various ways. If three-dimensional space is understood as a set of points , then a body is a subset of these points that fulfills certain properties.
In stereometry , a body is a restricted three-dimensional subset of three-dimensional space that is bounded on all sides by a finite number of flat or curved surface pieces, including these boundary surfaces. A set is called restricted if there is a correspondingly large sphere that completely encompasses the set. The union of the points of all the delimiting areas forms the surface of the body. The surface of a body divides space into two separate subsets, with the interior of the body being the subset that does not contain a straight line .
In geometric modeling , a body is a limited and regular subset of three-dimensional space. A quantity is called regular if it is the end of its interior. This condition ensures that a body also includes its edge and is completely three-dimensional, i.e. does not have any areas of lower dimensions. One speaks at this point of the homogeneity of a body. According to this definition, a body can also consist of several unconnected components.
The surface of a body can also consist of several unconnected parts. By assigning an orientation to these partial areas , a body can also be described using its surface. One then speaks of the surface representation ( boundary representation ) of the body.
Examples
The best-known bodies have flat or circular or spherical interfaces. Examples of solids in general are: cube , tetrahedron , pyramid , prism , octahedron , cylinder , cone , sphere and full torus .
Types of geometric bodies
polyhedron
A polyhedron is a geometric body whose boundaries are polygons . The most famous polyhedra include the regular polyhedra. These are the three-dimensional polygons delimited by regular polygons whose edges only point outwards and which are not infinitely large, such as the cube, the tetrahedron or the so-called soccer ball . There are only five types of these solids: the Platonic solids , which are dual with themselves or with one another, the Archimedean solids and the Catalan solids dual to them, and the Johnson solids . Then there are the prisms and the anti- prisms . There are only five regular polyhedra with which a complete alone chamber charge is possible: cubes, triangular and hexagonal prism twisted double wedge and truncated octahedron .
Convex body
If a geometric body is also convex , one speaks of a convex body. All regular polyhedra are convex. Convex bodies can also be derived from norms , for example the p-norms .
Body of revolution
Bodies, the surface of which is constructed by rotating a curve around a certain axis, are called bodies of revolution. Each cutting surface that is orthogonal to the axis of rotation has a circular or annular shape. These include spheres, cylinders, cones, truncated cones, torus and ellipsoid of revolution. The sphere occupies a special position because every straight line through its center is an axis of rotation.
additional
- Body nets , (physical) body models and software applications for dynamic spatial geometry and CAD are used to illustrate bodies .
- Geometry knows formulas for calculating the surface and volume of many bodies.
- Symmetry properties of individual bodies can be represented in group theory .
- Crystals are made up of (idealized) unit cells that can be understood as geometric bodies.
literature
- Tommy Bonnesen, W. Fennel: Theory of convex bodies . American Mathematical Soc., 1971, ISBN 0-8284-0054-7 .
Web links
Individual evidence
- ^ Walter Gellert, Herbert Kästner , Siegfried Neuber (eds.): Fachlexikon ABC Mathematik . Harri Deutsch, Thun / Frankfurt am Main 1998, ISBN 3-87144-336-0 , p. 298 .
- ^ Max K. Agoston: Computer Graphics and Geometric Modeling: Implementation & Algorithms . Springer, 2005, ISBN 1-84628-108-3 , pp. 158 .
- ↑ Leila de Floriani, Enrico Puppo: Representation and conversion issues in solid modeling . In: George Zobrist, CY Ho (Ed.): Intelligent Systems and Robotics . CRC Press, 2000, ISBN 90-5699-665-7 .