Solid geometry
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Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space).[1]
A solid figure is the
region of 3D space bounded by a two-dimensional surface; for example, a solid ball consists of a sphere and its interior
.
Solid geometry deals with the
cones (and truncated cones).[2]
History
The
regular solids, but the pyramid, prism, cone and cylinder were not studied until the Platonists. Eudoxus established their measurement, proving the pyramid and cone to have one-third the volume of a prism and cylinder on the same base and of the same height. He was probably also the discoverer of a proof that the volume enclosed by a sphere is proportional to the cube of its radius.[3]
Topics
Basic topics in solid geometry and stereometry include:
- lines
- dihedral angle and solid angle
- the cube, cuboid, parallelepiped
- the tetrahedron and other pyramids
- prisms
- octahedron, dodecahedron, icosahedron
- cylinders
- the sphere
- other quadrics: spheroid, ellipsoid, paraboloid and hyperboloids.
Advanced topics include:
- Desargues' theoremby using an extra dimension)
- further polyhedra
- descriptive geometry.
List of solid figures
Whereas a sphere is the surface of a ball, for other solid figures it is sometimes ambiguous whether the term refers to the surface of the figure or the volume enclosed therein, notably for a cylinder.
Figure | Definitions | Images |
---|---|---|
Parallelepiped |
|
|
Rhombohedron |
|
|
Cuboid |
|
|
Polyhedron | Flat polygonal faces, straight edges and sharp corners or vertices | |
Uniform polyhedron | vertex-transitive (i.e., there is an isometry mapping any vertex onto any other)
|
|
Prism | A corresponding sides of the two bases
|
|
Cone | Tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex | |
Cylinder | Straight parallel sides and a circular or oval cross section |
|
Ellipsoid | A surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation | |
Lemon | A lens (or less than half of a circular arc) rotated about an axis passing through the endpoints of the lens (or arc)[6] | |
Hyperboloid | A surface that is generated by rotating a hyperbola around one of its principal axes |
Techniques
Various techniques and tools are used in solid geometry. Among them,
vector techniques have a major impact by allowing the systematic use of linear equations and matrix
algebra, which are important for higher dimensions.
Applications
A major application of solid geometry and stereometry is in 3D computer graphics.
See also
Notes
- ^ The Britannica Guide to Geometry, Britannica Educational Publishing, 2010, pp. 67–68.
- ^ Kiselev 2008.
- 1911 Encyclopædia Britannica.
- ISBN 9780521277396.
- ^ Dupuis, Nathan Fellowes (1893). Elements of Synthetic Solid Geometry. Macmillan. p. 53. Retrieved December 1, 2018.
- ^ Weisstein, Eric W. "Lemon". Wolfram MathWorld. Retrieved 2019-11-04.
References
- Kiselev, A. P. (2008). Geometry. Vol. Book II. Stereometry. Translated by Givental, Alexander. Sumizdat.