Capsule (geometry)

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.

## Contents

* 1 Usages
* 2 Formulas
* 3 Generalization
* 4 Related shapes
* 5 References

## Usages

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

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

V=43πr3+(πr2h)=πr2(43r+h).

The surface area of a capsule of radius r whose cylinder part has height h is 2πr(2r+h).

## Generalization

A capsule can be equivalently described as the Minkowski sum of a ball of radius r with a line segment of length a.[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

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

## References

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.
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. ↑ 3.0 3.1 Frenkel, Daan (September 10, 1987). "Onsager's spherocylinders revisited". Journal of Physical Chemistry 91 (19): 4912–4916. doi:10.1021/j100303a008.
4. ↑ 4.0 4.1 Dzubiella, Joachim; Schmidt, Matthias (2000). "Topological defects in nematic droplets of hard spherocylinders". Physical Review E 62 (4): 5081–5091. doi:10.1103/PhysRevE.62.5081. PMID 11089056. Bibcode: 2000PhRvE..62.5081D.
5. ↑ 5.0 5.1 5.2 Pournin, Lionel; Weber, Mats; Tsukahara, Michel; Ferrez, Jean-Albert; Ramaioli, Marco; Liebling, Thomas M. (2005). "Three-dimensional distinct element simulation of spherocylinder crystallization". Granular Matter 7 (2–3): 119–126. doi:10.1007/s10035-004-0188-4. https://infoscience.epfl.ch/record/77620/files/10035_2004_Article_188.pdf.
6. ↑ "The effects of shape on the interaction of colloidal particles". Annals of the New York Academy of Sciences 51 (4): 627–659. May 1949. doi:10.1111/j.1749-6632.1949.tb27296.x.
7. ↑ 7.0 7.1 Pournin, Lionel; Liebling, Thomas M. (2005). "A generalization of Distinct Element Method to tridimensional particles with complex shapes". A.A. Balkema, Rotterdam. pp. 1375–1378. https://infoscience.epfl.ch/record/77616.
8. ↑ 8.0 8.1 Pournin, Lionel; Liebling, Thomas M. (2009). "From spheres to spheropolyhedra: generalized Distinct Element Methodology and algorithm analysis". 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.

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