Capsule (geometry)

From Wikipedia, the free encyclopedia
A two-dimensional orthographic projection at the left with a three-dimensional one at the right depicting a capsule

A capsule (from Latin capsula, "small box or chest"), or stadium of revolution, is a basic three-dimensional geometric shape consisting of a cylinder with hemispherical ends.[1] Another name for this shape is spherocylinder.[2][3][4][5]

It can also be referred to as an oval although the sides (either vertical or horizontal) are straight parallel.

Usages[edit]

The shape is used for some objects like containers for pressurised gases, windows of places like a jet, software buttons, building domes (like the U.S. Capitol, having the windows of the top hat that depict The Apotheosis of Washington inside designed with the appearance of the shape & placed in an omnidirectional pattern), mirrors, and pharmaceutical capsules.

In chemistry and physics, this shape is used as a basic model for non-spherical particles. It appears, in particular as a model for the molecules in liquid crystals[6][3][4] or for the particles in granular matter.[5][7][8]

Formulas[edit]

The volume of a capsule is calculated by adding the volume of a ball of radius (that accounts for the two hemispheres) to the volume of the cylindrical part. Hence, if the cylinder has height ,

.

The surface area of a capsule of radius whose cylinder part has height is .

Generalization[edit]

A capsule can be equivalently described as the Minkowski sum of a ball of radius with a line segment of length .[5] By this description, capsules can be straightforwardly generalized as Minkowski sums of a ball with a polyhedron. The resulting shape is called a spheropolyhedron.[7][8]

Related shapes[edit]

A capsule is the three-dimensional shape obtained by revolving the two-dimensional stadium around the line of symmetry that bisects the semicircles.

References[edit]

  1. ^ Sarkar, Dipankar; Halas, N. J. (1997). "General vector basis function solution of Maxwell's equations". Physical Review E. 56 (1, part B): 1102–1112. doi:10.1103/PhysRevE.56.1102. MR 1459098.
  2. ^ Kihara, Taro (1951). "The Second Virial Coefficient of Non-Spherical Molecules". Journal of the Physical Society of Japan. 6 (5): 289–296. doi:10.1143/JPSJ.6.289.
  3. ^ a b Frenkel, Daan (September 10, 1987). "Onsager's spherocylinders revisited". Journal of Physical Chemistry. 91 (19): 4912–4916. doi:10.1021/j100303a008. hdl:1874/8823. S2CID 96013495.
  4. ^ a b Dzubiella, Joachim; Schmidt, Matthias; Löwen, Hartmut (2000). "Topological defects in nematic droplets of hard spherocylinders". Physical Review E. 62 (4): 5081–5091. arXiv:cond-mat/9906388. Bibcode:2000PhRvE..62.5081D. doi:10.1103/PhysRevE.62.5081. PMID 11089056. S2CID 31381033.
  5. ^ a b c Pournin, Lionel; Weber, Mats; Tsukahara, Michel; Ferrez, Jean-Albert; Ramaioli, Marco; Liebling, Thomas M. (2005). "Three-dimensional distinct element simulation of spherocylinder crystallization" (PDF). Granular Matter. 7 (2–3): 119–126. doi:10.1007/s10035-004-0188-4.
  6. ^ Onsager, Lars (May 1949). "The effects of shape on the interaction of colloidal particles". Annals of the New York Academy of Sciences. 51 (4): 627–659. doi:10.1111/j.1749-6632.1949.tb27296.x. S2CID 84562683.
  7. ^ a b Pournin, Lionel; Liebling, Thomas M. (2005). "A generalization of Distinct Element Method to tridimensional particles with complex shapes". Powders and Grains 2005 Proceedings vol. II. A.A. Balkema, Rotterdam. pp. 1375–1378.
  8. ^ a b Pournin, Lionel; Liebling, Thomas M. (2009). "From spheres to spheropolyhedra: generalized Distinct Element Methodology and algorithm analysis". In Cook, William; Lovász, László; Vygen, Jens (eds.). Research Trends in Combinatorial Optimization. Springer, Berlin. pp. 347–363. doi:10.1007/978-3-540-76796-1_16. ISBN 978-3-540-76795-4.