Tube of a curve

Definition

Let $\gamma$ be a smooth curve (viz, a curve having a tangent at every point) in ${\mathbb{R}}^3$ and $r$ a length. The tube of radius $r$ about $\gamma$ is the union of the following circles:

For each point on the curve, choose the circle of radius $r$ about that point in the plane perpendicular to the tangent direction at that point.

$r$ is termed the tube radius.

For a closed curve that does not intersect itself, we can choose a tube radius such that no two of these circles described above intersect.

Note that when the curve is a planar curve, viz it lies completely in a plane, the tube of that curve is symmetric about the plane.

Examples

Tube of a straight line

The tube of a straight line is a right circular cylinder. The straight line is the axis of the cylinder.

Tube of a circle

The tube of a circle is a torus. The circle is the base circle of the torus.

Symmetries of the tube and the curve

In general, any symmetry of the curve also gives a symemtry of the tube. There may be nontrivial symmetries of the tube which are trivial on the curve. For instance, for a planar curve, the reflection about its plane is a nontrivial symmetry of the tube that is trivial on the curve.