Helix

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A helix (pl: helixes or helices ) is a type of Differentiable manifold|smooth space curve , i.e. a curve in three-dimensional space . It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis . Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for example, a spiral ramp – is called a helicoid . Helices are important in biology , as the DNA molecule is formed as double helix|two intertwined helices , and many protein s have helical substructures, known as alpha helix|alpha helices . The word helix comes from the Greek language|Greek word ???? , "twisted, curved"., Henry George Liddell, Robert Scott, A Greek-English Lexicon , on Perseus

Types

Helices can be either right-handed or left-handed. With the line of sight along the helix's axis, if a clockwise screwing motion ???:Elica-conica.gifmoves 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 (mathematics)|chirality ) is a property of the helix, not of the perspective: a right-handed helix cannot be turned or flipped to look like a left-handed one unless it is viewed in a mirror, and vice versa.

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

The pitch of a helix is the width 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."" by Sándor Kabai, Wolfram Demonstrations Project .

A conic helix 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. An example is the Corkscrew (Cedar Point)|Corkscrew roller coaster at Cedar Point amusement park.

A circular helix, (i.e. one with constant radius) has constant band curvature and constant Torsion of curves|torsion .

A curve is called a general helix or cylindrical helix O'Neill, B. Elementary Differential Geometry, 1961 pg 72 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 of curves|torsion is constant.O'Neill, B. Elementary Differential Geometry, 1961 pg 74

Mathematical description

In mathematics , a helix is a Differential geometry of curves|curve in 3- dimension al space. The following Parametric equation|parametrisation in Cartesian coordinate system|Cartesian coordinates defines a helix:

:

x(t) = \cos(t),\,

:

y(t) = \sin(t),\,

:

z(t) = t.\,

As the parameter t increases, the point ( x ( t ), y ( t ), z ( t )) traces a right-handed helix of pitch 2 pi|p and radius 1 about the z -axis, in a right-handed coordinate system.

In cylindrical coordinates ( r , ? , h ), the same helix is parametrised by:
:

r(t) = 1,\,

:

\theta(t) = t,\,

:

h(t) = t.\,

A circular helix of radius a and pitch 2 pb is described by the following parametrisation:

:

x(t) = a\cos(t),\,

:

y(t) = a\sin(t),\,

:

z(t) = bt.\,

Another way of mathematically constructing a helix is to plot a complex valued exponential function ( e ^xi) taking imaginary arguments (see Euler's formula ).
Except for rotation s, translation (geometry)|translation s, 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

The length of a circular helix of radius a and pitch 2 pb expressed in rectangular coordinates as
:

t\mapsto (a\cos t, a\sin t, bt), t\in

equals

T\cdot \sqrt{a^2+b^2}

, its Curvature#One dimension in three dimensions: Curvature of space curves|curvature is

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

and its Torsion of a curve|torsion is

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

Examples

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 .

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