Английская Википедия:Helix

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Файл:Helix.svg

The right-handed helix Шаблон:Math for Шаблон:Math with arrowheads showing direction of increasing Шаблон:Mvar

A helix (Шаблон:IPAc-en; Шаблон:Plural form) is a shape like a corkscrew. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word helix comes from the Greek word Шаблон:Lang, "twisted, curved". A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called a helicoid.

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.

A circular helix (i.e. one with constant radius) has constant band curvature and constant torsion.

A conic helix, also known as a conic spiral, may be defined as a spiral 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 a general helix or cylindrical helix 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 of curvature to torsion is constant.

A curve is called a slant helix if its principal normal makes a constant angle with a fixed line in space. It can be constructed by applying a transformation to the moving frame of a general helix.

For more general helix-like space curves can be found, see space spiral; e.g., spherical spiral.

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.

Файл:Two Types of Helix.svg

Two types of helix shown in comparison. This shows the two chiralities 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 the perspective (view-angle)

Mathematical description

Файл:Rising circular.gif

A helix composed of sinusoidal Шаблон:Mvar and Шаблон:Mvar components

In mathematics, a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a particular helix; perhaps the simplest equations for one is

<math>\begin{align}

x(t) &= \cos(t),\\ y(t) &= \sin(t),\\ z(t) &= t. \end{align}</math>

As the parameter Шаблон:Mvar increases, the point Шаблон:Math traces a right-handed helix of pitch Шаблон:Math (or slope 1) and radius 1 about the Шаблон:Mvar-axis, in a right-handed coordinate system.

In cylindrical coordinates Шаблон:Math, the same helix is parametrised by:

<math>\begin{align}

r(t) &= 1,\\ \theta(t) &= t,\\ h(t) &= t. \end{align}</math>

A circular helix of radius Шаблон:Mvar and slope Шаблон:Math (or pitch Шаблон:Math) is described by the following parametrisation:

<math>\begin{align}

x(t) &= a\cos(t),\\ y(t) &= a\sin(t),\\ z(t) &= bt. \end{align}</math>

Another way of mathematically constructing a helix is to plot the complex-valued function Шаблон:Math as a function of the real number Шаблон:Mvar (see Euler's formula). The value of Шаблон:Mvar 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 Шаблон:Mvar, Шаблон:Mvar or Шаблон:Mvar components.

Arc length, curvature and torsionШаблон:Anchor

A circular helix of radius Шаблон:Mvar and slope Шаблон:Math (or pitch Шаблон:Math) expressed in Cartesian coordinates as

<math>t\mapsto (a\cos t, a\sin t, bt), t\in [0,T]</math>

has an arc length of

<math>A = T\cdot \sqrt{a^2+b^2},</math>

a curvature of

<math>\frac{|a|}{a^2+b^2},</math>

and a torsion of

<math>\frac{b}{a^2+b^2}.</math>

A helix has constant non-zero curvature and torsion.

A helix is the vector-valued function

<math display="block">\begin{align} \mathbf{r}&=a\cos t \mathbf{i}+a\sin t \mathbf{j}+ b t\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{align}</math>

So a helix can be reparameterized as a function of Шаблон:Mvar, which must be unit-speed:

<math display="block">\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}</math>

The unit tangent vector is

<math display="block">\frac{d \mathbf{r}}{d s} = \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}</math>

The normal vector is

<math display="block">\frac{d \mathbf{T}}{d s} = \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}</math>

Its curvature is

<math display="block">\kappa = \left|\frac{d\mathbf{T}}{ds}\right|= \frac{|a|}{a^2 +b^2 }</math>.

The unit normal vector is

<math display="block">\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}</math>

The binormal vector is

<math display="block"> \begin{align} \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{align}</math>

Its torsion is <math display="block">\tau = \left| \frac{d\mathbf{B}}{ds} \right| = \frac{b}{a^2 +b^2}.</math>

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 screw threads are right-handed helices. The alpha helix in biology as well as the A and B forms of DNA are also right-handed helices. The Z form of DNA is left-handed.

In music, pitch space is often modeled with helices or double helices, most often extending out of a circle such as the circle of fifths, so as to represent octave 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 a folded molecular helix reported by Lehn et al.[9]

  Crystal structure of a folded molecular helix reported by Lehn et al.

• A natural left-handed helix, made by a climber plant

  A natural left-handed helix, made by a climber plant

• A charged particle in a uniform magnetic field following a helical path

  A charged particle in a uniform magnetic field following a helical path

• A helical coil spring

  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