Solid geometry

In mathematics, solid geometry is the traditional name for the geometry of three-dimensional Euclidean space (i.e., 3D geometry).

Hyperboloid of one sheet

Stereometry deals with the measurements of volumes of various solid figures (three-dimensional figures), including pyramids, prisms and other polyhedrons; cylinders; cones; truncated cones; and balls bounded by spheres.[1]

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.[2]

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.

Solid figures

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.

Figure	Definitions	Images
Parallelepiped	A polyhedron with six faces (hexahedron), each of which is a parallelogram A hexahedron with three pairs of parallel faces A prism of which the base is a parallelogram
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[3] Some sources also require that each of the faces is a rectangle (so each pair of adjacent faces meets in a right angle). 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.[4]
Polyhedron	Flat polygonal faces, straight edges and sharp corners or vertices	Small stellated dodecahedron Toroidal polyhedron
Uniform polyhedron	Regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other)	Tetrahedron Dodecahedron
Prism	A polyhedron comprising an n-sided polygonal base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces (necessarily all parallelograms) joining 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	A right circular cone and an oblique circular cone
Cylinder	Straight parallel sides and a circular or oval cross section	A solid elliptic cylinder A right and an oblique circular cylinder
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	Examples of ellipsoids with equation ${x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1:$ sphere (top, a=b=c=4), spheroid (bottom left, a=b=5, c=3), tri-axial ellipsoid (bottom right, a=4.5, b=6, c=3)
Lemon	A lens (or less than half of a circular arc) rotated about an axis passing through the endpoints of the lens (or arc)[5]
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, 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.

gollark: Surely if it was this easy to manipulate large amounts of people into weird political views someone would already be doing it.

gollark: I am still satisfied with the IPS displays on my laptop/phone for now.

gollark: General COVID-19 breakage and a shortage of drivers because brexit.

gollark: Oh, and unrelatedly, apparently UK food supply chains are very close to failure (it is already bad enough that there are significant gaps on shelves). Fun!

gollark: Why? We could replace the important ones.

Notes

Kiselev 2008.
...paraphrased and taken in part from the 1911 Encyclopædia Britannica.
Robertson, Stewart Alexander (1984). Polytopes and Symmetry. Cambridge University Press. p. 75. 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. Book II. Stereometry. Translated by Givental, Alexander. Sumizdat.CS1 maint: ref=harv (link)

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[1] Kiselev 2008.

[2] ...paraphrased and taken in part from the 1911 Encyclopædia Britannica.

[3] Robertson, Stewart Alexander (1984). Polytopes and Symmetry. Cambridge University Press. p. 75. ISBN 9780521277396.

[4] Dupuis, Nathan Fellowes (1893). Elements of Synthetic Solid Geometry. Macmillan. p. 53. Retrieved December 1, 2018.

[mathworld-5] Weisstein, Eric W. "Lemon". Wolfram MathWorld. Retrieved 2019-11-04.