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Tube with a horopter curve as bore

Synonyms: canal surface, channel surface.

Parametrization:  where  and  are the normal binormal vectors of the spine curve (G0) 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:
Here is, for example, a tube with square section: be careful to eliminate the torsion in order to avoid the case on the right!
2) take a circle with variable radius, still orthogonal to the spine curve: we get the notion of tube with variable section.
  - 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.
Here is a cross-section view of a tube with variable section generated by a circle centered on a circle and passing by a line. The figure on the right shows that the spheres centered on the circle and tangent to the line are not tangent to the other curve...

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