Solid geometry
Field of mathematics dealing with three-dimensional Euclidean spaces
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Solid geometry deals with the measurements of volumes of various solids, including pyramids, prisms, cubes (and other polyhedrons), cylinders, cones (including truncated) and other solids of revolution.
History
The Pythagoreans dealt with 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.
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Topics
Basic topics in solid geometry and stereometry include:
- incidence of planes and lines
- dihedral angle and solid angle
- the cube, cuboid, parallelepiped
- the tetrahedron and other pyramids
- prisms
- octahedron, dodecahedron, icosahedron
- cones and cylinders
- the sphere
- other quadrics: spheroid, ellipsoid, paraboloid and hyperboloids.
Advanced topics include:
- projective geometry of three dimensions (leading to a proof of Desargues' theorem by 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.
Techniques
Various techniques and tools are used in solid geometry. Among them, analytic geometry and 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.