Right circular cylinder

Definition

A right circular cylinder, or more precisely, a right circular cylindrical surface, is a union of all circles centered at points on a given line, of fixed radius, and each circle in the plane perpendicular to the line at that point. The line is termed the axis of the cylinder.

The above is the infinite right circular cylindrical surface. The finite version is obtained by replacing the line with an open line segment.

The term solid cylinder is used to refer to the unino of discs, instead of circles. The cylindrical surface is then termed the lateral surface of the solid cylinder.

Terminology

• The line comprising centers is termed the axis of the cylinder.
• Planes perpendicular to the axis are termed cross-sectional planes
• The equal radius of all circles is termed the radius or base radius of the cylinder and is denoted by $r$
• For a finite cylinder, the length of the open line segment is termed the height of the cylinder and is denoted by $h$

Equational descriptions

Cartesian parametric description

If the axis is the $z$-axis, the parametric description is in terms of $t\in \mathbb {R}$ and $\theta$ an angle.

$x=r\cos \theta ,y=r\sin \theta ,z=t$

For a finite cylinder, we have, instead of $t\in \mathbb {R}$, $t\in [t_{0},t_{1}]$ where $t_{1}-t_{0}=h$.

Cartesian equational description

If the axis is the $z$-axis, the equational description in Cartesian coordinates is:

$x^{2}+y^{2}=r^{2}$

In other words, the $z$-coordinate is arbitrary, and any section with the $xy$-plane looks like a circle of radius $r$ centered at the origin.

Structure and symmetry

The infinite right circular cylinder is topologically, differentially, and geometrically the direct product of the circle $S^{1}$ and the real line $\mathbb {R}$. Moreover, this is an isometry-invariant factorization, in the sense that any isometry must take cross-sectional circles to cross-sectional circles, and lines parallel to the axis, to lines parallel to the axis.

Using this, we can compute that the group of isometries of the right circular cylinder is the semidirect product of the group itself (acting via the regular representation) and the Klein-four group, comprising the following four operations:

• The identity element
• Reflections about the $xy$-plane
• Rotation by an angle of $\pi$ about the $z$-axis
• Composite of the above two

Curvatures

The two principal directions at any point are the direction along the real line and along the circle. Along the real line, the curvature is 0, while along the circle, the curvature is the usual curvature associated with a circle. Thus, the Gaussian curvature at every point is zero.

Properties

Since there is an underlying group structure compatible with the metric, there is an isometry taking any point to any other point. Thus, the right circular cylider is homogeneous.

This is on account of the Gaussian curvature (the two-dimensional particular case of sectional curvature) being zero at every point.