Solid geometry


In mathematics, solid geometry is the traditional name for the geometry of three-dimensional Euclidean space.
Stereometry deals with the measurements of volumes of various solid figures, including pyramids, prisms and other polyhedrons; cylinders; cones; truncated cones; and balls bounded by spheres.

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.

Topics

Basic topics in solid geometry and stereometry include:
Advanced topics include:
Whereas a sphere is the surface of a ball, it is sometimes ambiguous whether the term refers to the surface of the figure or the volume enclosed therein, notably for a cylinder. The following table includes major types of shapes that either constitute or define a volume.
FigureDefinitionsImages-
Parallelepiped
-
Rhombohedron
  • A parallelepiped where all edges are the same length
  • A cube, except that its faces are not squares but rhombi
    		• -
    Cuboid
  • A convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube
  • Some sources also require that each of the faces is a rectangle. This more restrictive type of cuboid is also known as a rectangular cuboid, right cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or rectangular parallelepiped.
    		• -
    PolyhedronFlat polygonal faces, straight edges and sharp corners or vertices-
    Uniform polyhedronRegular polygons as faces and is vertex-transitive -
    PrismA polyhedron comprising an n-sided polygonal base, a second base which is a translated copy of the first, and n other faces joining corresponding sides of the two bases-
    ConeTapers smoothly from a flat base to a point called the apex or vertex-
    CylinderStraight parallel sides and a circular or oval cross section-
    EllipsoidA surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation,
    spheroid,

    tri-axial'' ellipsoid     		-
    LemonA lens rotated about an axis passing through the endpoints of the lens -
    HyperboloidA 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, 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.