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TUBE or TUBULAR SURFACE or PIPE
Tube with a horopter
curve as bore
Synonyms: canal surface, channel surface. 
Parametrization:
where
and are
the normal binormal vectors of the spine curve (G_{0})
and a the radius of the tube.
Area of the section : . Volume of this section: . 
The tubes with spine curve (or bore)
the curve
are the circled surfaces generated
by a circle with constant radius centered on
and the plane of which is always normal to this curve.
They are also the envelopes
of a sphere with constant radius centered on .
They are the Monge surfaces
with circular generatrix.
The visible
outline of a tube is composed of two parallel
curves of the projection of the spine curve.
Examples: the sphere (case where
is reduced to a point), the cylinder
of revolution, the torus, the
coil.
The notion can be generalized to three directions:
1) take a non circular section: we get the notion of Monge surface, with a closed generatrix: 

2) take a circle with variable radius, still orthogonal
to the spine curve: we get the notion of tube with variable section.
Examples:  the surfaces of revolution  the sine tori of the second kind. 
Here is, for example, a tube the section of which varies sinusoidally. 
3) take spheres with variable radius centered
on , and
consider their envelope.
When is linear, the notions 2) and 3) coincide, but not in the general case (cf. opposite). It is this general notion of envelope of spheres with variable radius that is sometimes referred to as "canal surface" [gray]. The characterization is: circled surface the circles of which are radii of curvature. Examples: the Dupin cyclides. 

See also the solenoids,
coiling of a wire around a tube.
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© Robert FERRÉOL 2017