Helix
A helix (
Properties and types
The pitch of a helix is the height of one complete helix turn, measured parallel to the axis of the helix.
A double helix consists of two (typically congruent) helices with the same axis, differing by a translation along the axis.[3]
A circular helix (i.e. one with constant radius) has constant band
A
A curve is called a general helix or cylindrical helix
A curve is called a slant helix if its principal normal makes a constant angle with a fixed line in space.[6] It can be constructed by applying a transformation to the moving frame of a general helix.[7]
For more general helix-like space curves can be found, see
Handedness
Helices can be either right-handed or left-handed. With the line of sight along the helix's axis, if a clockwise screwing motion moves the helix away from the observer, then it is called a right-handed helix; if towards the observer, then it is a left-handed helix. Handedness (or chirality) is a property of the helix, not of the perspective: a right-handed helix cannot be turned to look like a left-handed one unless it is viewed in a mirror, and vice versa.
Mathematical description
In
As the parameter t increases, the point (x(t),y(t),z(t)) traces a right-handed helix of pitch 2π (or slope 1) and radius 1 about the z-axis, in a right-handed coordinate system.
In
A circular helix of radius a and slope a/b (or pitch 2πb) is described by the following parametrisation:
Another way of mathematically constructing a helix is to plot the complex-valued function exi as a function of the real number x (see Euler's formula). The value of x and the real and imaginary parts of the function value give this plot three real dimensions.
Except for rotations, translations, and changes of scale, all right-handed helices are equivalent to the helix defined above. The equivalent left-handed helix can be constructed in a number of ways, the simplest being to negate any one of the x, y or z components.
Arc length, curvature and torsion
A circular helix of radius a and slope a/b (or pitch 2πb) expressed in Cartesian coordinates as
has an arc length of
a curvature of
and a torsion of
A helix has constant non-zero curvature and torsion.
A helix is the vector-valued function
So a helix can be reparameterized as a function of s, which must be unit-speed:
The unit tangent vector is
The normal vector is
Its curvature is
The unit normal vector is
The binormal vector is
Its torsion is
Examples
An example of a double helix in molecular biology is the nucleic acid double helix.
An example of a conic helix is the Corkscrew roller coaster at Cedar Point amusement park.
Some curves found in nature consist of multiple helices of different handedness joined together by transitions known as tendril perversions.
Most hardware
In
In aviation, geometric pitch is the distance an element of an airplane propeller would advance in one revolution if it were moving along a helix having an angle equal to that between the chord of the element and a plane perpendicular to the propeller axis; see also:
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A natural left-handed helix, made by a climber plant
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A charged particle in a uniform magnetic field following a helical path
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A helical coil spring
See also
- Alpha helix
- Arc spring
- Boerdijk–Coxeter helix
- Circular polarization
- Collagen helix
- Helical symmetry
- Helicity
- Helix angle
- Helical axis
- Hemihelix
- Seashell surface
- Solenoid
- Superhelix
- Triple helix
References
- ^ ἕλιξ Archived 2012-10-16 at the Wayback Machine, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
- ^ Weisstein, Eric W. "Helicoid". MathWorld.
- ^ "Double Helix Archived 2008-04-30 at the Wayback Machine" by Sándor Kabai, Wolfram Demonstrations Project.
- ^ O'Neill, B. Elementary Differential Geometry, 1961 pg 72
- ^ O'Neill, B. Elementary Differential Geometry, 1961 pg 74
- ^ Izumiya, S. and Takeuchi, N. (2004) New special curves and developable surfaces. Turk J Math Archived 2016-03-04 at the Wayback Machine, 28:153–163.
- ^ Menninger, T. (2013), An Explicit Parametrization of the Frenet Apparatus of the Slant Helix. arXiv:1302.3175 Archived 2018-02-05 at the Wayback Machine.
- ^ Weisstein, Eric W. "Helix". MathWorld.
- .