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CYLINDRICAL HELIX
lefthanded helix 
righthanded helix 
First study (that did not reach our times) credited to
Apollonius of Perga (2nd century BC) and second study by Geminus of Rhodes
(1st century BC).
Other name: circular helix. 
Cartesian parametrization: .
System of cylindrical equations: . Curvilinear abscissa: , where . Radius of curvature: , angle of curvature: at / c. Radius of torsion: , angle of torsion: bt / c. Length of a coil: . Center of curvature: (describes another cylindrical helix). 
The cylindrical helix can be defined as a helix
traced on a vertical cylinder
of revolution, or a rhumb
line of this cylinder (i.e., in both cases, a curve forming a constant
angle with respect to the axis of the cylinder), or a geodesic
of this cylinder (in other words, a curve that becomes a line when the
cylinder is developed) or a solenoid
with linear bore.
Intrinsic characterization: constant curvature and torsion.
The radius of the helix is a, and its shift is (it is the distance between two consecutive coils) and b is sometimes called reduced shift of the helix. The angle of the helix is the constant angle (equal to ) formed by its tangent with respect to any plane orthogonal to Oz. The helix is righthanded when e = 1 (it “goes up” counterclockwise and an observer located outside of it sees it going up from left to right) and lefthanded when e =  1 (it “goes up” clockwise).
The DNA spirals along a double righthanded helix. 


Finally, in Brazil, a species of bat flying invariably along righthanded helices has been observed.
The trajectories of a charge placed in a uniform magnetic
field and subject to the Laplace force is a cylindrical
helix with axis parallel to that of the field (or a circle if the velocity
is perpendicular to the field, a straight line if it is parallel). Same
result for an inelastic flexible wire carrying a continuous electric current
placed in the field (and not subject to gravity). The shape assumed by
this wire was called by Riebke "electrodynamic catenary", and is in fact
a cylindrical helix, with the same limit case as before.
Proof of this fact:
Newton's second law and the expression of the Laplace force give: (1) where is the tension of the wire, the tangent unit vector, the magnetic field and I the intensity of the current; shows that the norm of the tension is constant: T = constant. (1) can be therefore be written ; shows that : the tangent forms a constant angle with respect to a fixed direction, the curve is a helix. The first Frenet formula and (2) yield: the radius of curvature is constant. The radius of torsion of the helix, classically given by is therefore itself also constant: the curve is a cylindrical helix with axis parallel to the field lines. 
Nota: why are the bladebased propelling systems for boats or planes called helices? It is because when the boat moves with constant speed, the motion of the helix with respect to a fixed frame is a helicoidal motion, and all its points describe cylindrical helices.
For the projections of the cylindrical helix, see at trochoid,
cochleoid
and
hyperbolic
spiral.
See also the helicoids,
the other cylindrical curves,
the coil (tube
with a cylindricalhelixshaped bore), and the revolution
of the sinusoid.
More beautiful images by Amain Juhel.
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© Robert FERRÉOL, Jacques MANDONNET 2018