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 perpendicular to the tangent vector , and the circle is in the plane 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

(1)

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.

Equivalently, a tube is the constant-radius case of a canal surface. The envelope of the spheres centered at with radius satisfies

			(2)
			(3)

The second equation makes each cross-sectional circle perpendicular to the tangent vector. For a closed embedded spine curve and sufficiently small , this envelope is the surface at distance from the curve. At larger radii the envelope can self-intersect and need not coincide with that distance surface.