spherocylinders

A spherocylinder, also called a capsule, is a three-dimensional convex body formed by a cylinder of radius R and height L with hemispherical end-caps of the same radius R. The total length of the shape is L + 2R and the diameter is 2R. In geometry it is the Minkowski sum of a line segment of length L and a sphere of radius R. The cylindrical portion and the hemispherical caps meet smoothly, yielding a continuous surface.

Key dimensions and formulas:

- Radius: R

- Cylindrical length: L

- Total length: L + 2R

- Diameter: 2R

- Volume: V = πR^2L + (4/3)πR^3

- Surface area: S = 2πRL + 4πR^2

A common convention defines the aspect ratio as L/D = L/(2R), although some texts use L/R.

Special cases:

- If L = 0, the shape reduces to a sphere of radius R.

- If R → 0, the spherocylinder degenerates toward a line segment of length L.

Applications and context:

Spherocylinders serve as simple models for elongated particles in physics and materials science, including liquid crystals and colloidal suspensions. They are used in theoretical studies (e.g., Onsager-type theories of nematic ordering), numerical simulations of hard-body packings, and computer graphics as efficient collision-detection capsules. Due to their combination of a cylindrical body with rounded ends, they provide a tractable yet representative geometry for exploring anisotropy, packing, and dynamical behavior of rod-like particles.