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Helix

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From Wikipedia, the free encyclopedia

Space curve that winds around a line

For other uses, seeHelix (disambiguation).

(l-r) Tension, compression and torsion coil springs

The right-handed helix(cost, sint,t) for0 ≤t ≤ 4π with arrowheads showing direction of increasingt

Ahelix (/ˈhiːlɪks/;pl. helices) is a shape like a cylindrical coil spring or the thread of amachine screw. It is a type ofsmooth skew curve withtangent lines at a constantangle to a fixed axis. Helices are important inbiology, as theDNA molecule is formed astwo intertwined helices, and manyproteins have helical substructures, known asalpha helices. The wordhelix comes from theGreek wordἕλιξ, "twisted, curved".^[1] A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called ahelicoid.^[2]

Properties and types

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Thepitch of a helix is the height of one complete helixturn, measured parallel to the axis of the helix.

Adouble helix consists of two (typicallycongruent) helices with the same axis, differing by a translation along the axis.^[3]

Acircular helix (i.e. one with constant radius) has constant bandcurvature and constanttorsion. The slope of a circular helix is commonly defined as the ratio of the circumference of the circular cylinder that it spirals around, and its pitch (the height of one complete helix turn).

Aconic helix, also known as aconic spiral, may be defined as aspiral on a conic surface, with the distance to the apex an exponential function of the angle indicating direction from the axis.

A curve is called ageneral helix orcylindrical helix^[4] if its tangent makes a constant angle with a fixed line in space. A curve is a general helix if and only if the ratio ofcurvature totorsion is constant.^[5]

A curve is called aslant 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, seespace spiral; e.g.,spherical spiral.

Handedness

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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 (orchirality) 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.

**Two types of helix shown in comparison**. This shows the twochiralities of helices. One is left-handed and the other is right-handed. Each row compares the two helices from a different perspective. The chirality is a property of the object, not of theperspective (view-angle)

Mathematical description

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A helix composed of sinusoidalx andy components

Inmathematics, a helix is acurve in 3-dimensional space. The followingparametrisation inCartesian coordinates defines a particular helix;^[8] perhaps the simplest equations for one is

{\begin{aligned}x(t)&=\cos(t),\\y(t)&=\sin(t),\\z(t)&=t.\end{aligned}}

As theparametert increases, the point $(x(t),y(t),z(t))$ traces a right-handed helix of pitch2π (or slope 1) and radius 1 about thez-axis, in a right-handed coordinate system.

Incylindrical coordinates(r,θ,h), the same helix is parametrised by:

{\begin{aligned}r(t)&=1,\\\theta (t)&=t,\\h(t)&=t.\end{aligned}}

A circular helix of radiusa and slope⁠a/b⁠ (or pitch2πb) is described by the following parametrisation:

{\begin{aligned}x(t)&=a\cos(t),\\y(t)&=a\sin(t),\\z(t)&=bt.\end{aligned}}

Another way of mathematically constructing a helix is to plot the complex-valued functione^xi as a function of the real numberx (seeEuler's formula).The value ofx and the real and imaginary parts of the function value give this plot three real dimensions.

Except forrotations,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 thex,y orz components.

Arc length, curvature and torsion

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A circular helix of radius $a>0$ and slope⁠a/b⁠ (or pitch2πb) expressed in Cartesian coordinates as theparametric equation

t\mapsto (a\cos t,a\sin t,bt),t\in [0,T]

has anarc length of

A=T\cdot {\sqrt {a^{2}+b^{2}}},

acurvature of

{\frac {a}{a^{2}+b^{2}}},

and atorsion of

{\frac {b}{a^{2}+b^{2}}}.

Arc length per revolution ( $T=2\pi$ ): $A={\sqrt {(2\pi a)^{2}+(2\pi b)^{2}}}$ or $A={\sqrt {(2\pi a)^{2}+p^{2}}}$ where $p=$ pitch.

Twisted length per unit straight length (axial length): $A={\frac {\sqrt {(2\pi a)^{2}+p^{2}}}{p}}$

A helix has constant non-zero curvature and torsion.

A helix is the vector-valued function

${\begin{aligned}\mathbf {r} &=a\cos t\mathbf {i} +a\sin t\mathbf {j} +bt\mathbf {k} \\[6px]\mathbf {v} &=-a\sin t\mathbf {i} +a\cos t\mathbf {j} +b\mathbf {k} \\[6px]\mathbf {a} &=-a\cos t\mathbf {i} -a\sin t\mathbf {j} +0\mathbf {k} \\[6px]|\mathbf {v} |&={\sqrt {(-a\sin t)^{2}+(a\cos t)^{2}+b^{2}}}={\sqrt {a^{2}+b^{2}}}\\[6px]|\mathbf {a} |&={\sqrt {(-a\sin t)^{2}+(a\cos t)^{2}}}=a\\[6px]s(t)&=\int _{0}^{t}{\sqrt {a^{2}+b^{2}}}d\tau ={\sqrt {a^{2}+b^{2}}}t\end{aligned}}$

So a helix can be reparameterized as a function ofs, which must be unit-speed:

$\mathbf {r} (s)=a\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +a\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +{\frac {bs}{\sqrt {a^{2}+b^{2}}}}\mathbf {k}$

The unit tangent vector is

${\frac {d\mathbf {r} }{ds}}=\mathbf {T} ={\frac {-a}{\sqrt {a^{2}+b^{2}}}}\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +{\frac {a}{\sqrt {a^{2}+b^{2}}}}\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +{\frac {b}{\sqrt {a^{2}+b^{2}}}}\mathbf {k}$

The normal vector is

${\frac {d\mathbf {T} }{ds}}=\kappa \mathbf {N} ={\frac {-a}{a^{2}+b^{2}}}\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +{\frac {-a}{a^{2}+b^{2}}}\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +0\mathbf {k}$

Its curvature is

$\kappa =\left|{\frac {d\mathbf {T} }{ds}}\right|={\frac {a}{a^{2}+b^{2}}}$ .

The unit normal vector is

$\mathbf {N} =-\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} -\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +0\mathbf {k}$

The binormal vector is

${\begin{aligned}\mathbf {B} =\mathbf {T} \times \mathbf {N} &={\frac {1}{\sqrt {a^{2}+b^{2}}}}\left(b\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} -b\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +a\mathbf {k} \right)\\[12px]{\frac {d\mathbf {B} }{ds}}&={\frac {1}{a^{2}+b^{2}}}\left(b\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +b\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +0\mathbf {k} \right)\end{aligned}}$

Its torsion is $\tau =\left|{\frac {d\mathbf {B} }{ds}}\right|={\frac {b}{a^{2}+b^{2}}}.$

Examples

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An example of a double helix in molecular biology is thenucleic acid double helix.

An example of a conic helix is theCorkscrew roller coaster atCedar Point amusement park.

Some curves found in nature consist of multiple helices of different handedness joined together by transitions known astendril perversions.

Most hardwarescrew threads are right-handed helices. The alpha helix in biology as well as theA andB forms of DNA are also right-handed helices. TheZ form of DNA is left-handed.

Inmusic,pitch space is often modeled with helices or double helices, most often extending out of a circle such as thecircle of fifths, so as to representoctave equivalency.

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:pitch angle (aviation).

Crystal structure of afolded molecular helix reported byLehnet al.^[9]
A natural left-handed helix, made by aclimber plant
A charged particle in a uniformmagnetic field following a helical path
A helicalcoil spring
ADNA double helix

References

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^ἕλιξ Archived 2012-10-16 at theWayback Machine, Henry George Liddell, Robert Scott,A Greek-English Lexicon, on Perseus
^Weisstein, Eric W."Helicoid".MathWorld.
^"Double Helix Archived 2008-04-30 at theWayback 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 theWayback 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 theWayback Machine.
^Weisstein, Eric W."Helix".MathWorld.
^Schmitt, J.-L.; Stadler, A.-M.; Kyritsakas, N.; Lehn, J.-M. (2003). "Helicity-Encoded Molecular Strands: Efficient Access by the Hydrazone Route and Structural Features".Helvetica Chimica Acta.86 (5):1598–1624.Bibcode:2003HChAc..86.1598S.doi:10.1002/hlca.200390137.