Tube

A tube of radius of a set is the set of points at a distance from . In particular, if is a regular space curve whose curvature does not vanish, then the normal vector and binormal vector are always perpendicular to , and the circle is perpendicular to at . So as the circle moves around , it traces out a tube, provided the tube radius is small enough so that the tube is not self-intersecting. A formula for the tube around a curve is therefore given by

for over the range of the curve and . The illustrations above show tubes corresponding to a circle, helix, and two torus knots.

The surface generated by constructing a tube around a circle is known as a torus.