Helix

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The right-handed helix (cos t, sin t, t) from t = 0 to 4p with arrowheads showing direction of increasing t Helix.svg
The right-handed helix (cos t, sin t, t) from t = 0 to 4π with arrowheads showing direction of increasing t

A helix ( /ˈhlɪks/ ), plural helixes or helices ( /ˈhɛlɪsz/ ), is a shape like a corkscrew or spiral staircase. 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 ἕλιξ, "twisted, curved". [1] A "filled-in" helix – for example, a "spiral" (helical) ramp – is called a helicoid. [2]

Contents

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 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.

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) 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)

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.

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 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 roller coaster at Cedar Point amusement park.

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

A curve is called a general helix or cylindrical 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 of curvature to torsion is constant. [5]

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.

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]

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

Mathematical description

A helix composed of sinusoidal x and y components Rising circular.gif
A helix composed of sinusoidal x and y components

In mathematics, a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a particular helix; [8] perhaps the simplest equations for one is

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

A circular helix of radius a and slope b/a (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

The length of a circular helix of radius a and slope b/a (or pitch 2πb) expressed in rectangular coordinates as

equals , its curvature is and its torsion is 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 , 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

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.

See also

Related Research Articles

A centripetal force is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.

Ellipse Plane curve: conic section

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity , a number ranging from to .

Curvature Measure of the property of a curve or a surface to be "bended"

In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

Law of sines

In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle to the sines of its angles. According to the law,

Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites and crystallographic texture analysis.

Parametric equation Representation of a curve by a function of a parameter

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization of the object.

Frenet–Serret formulas

In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a continuous, differentiable curve in three-dimensional Euclidean space ℝ3, or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the derivatives of the so-called tangent, normal, and binormal unit vectors in terms of each other. The formulas are named after the two French mathematicians who independently discovered them: Jean Frédéric Frenet, in his thesis of 1847, and Joseph Alfred Serret in 1851. Vector notation and linear algebra currently used to write these formulas were not yet in use at the time of their discovery.

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.

Arc length Distance along a curve

Arc length is the distance between two points along a section of a curve.

In the elementary differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting out of the plane of curvature. Taken together, the curvature and the torsion of a space curve are analogous to the curvature of a plane curve. For example, they are coefficients in the system of differential equations for the Frenet frame given by the Frenet–Serret formulas.

Theoretical motivation for general relativity

A theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation a priori. This provides a means to inform and verify the formalism.

Torsion tensor

In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves. In the geometry of surfaces, the geodesic torsion describes how a surface twists about a curve on the surface. The companion notion of curvature measures how moving frames "roll" along a curve "without twisting".

Elliptic coordinate system

In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the -axis of the Cartesian coordinate system.

Elliptic cylindrical coordinates

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular -direction. Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the -axis of the Cartesian coordinate system.

In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

Radius of curvature

In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof.

Boerdijk–Coxeter helix Linear stacking of regular tetrahedra that form helices

The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices. There are two chiral forms, with either clockwise or counterclockwise windings. Unlike any other stacking of Platonic solids, the Boerdijk–Coxeter helix is not rotationally repetitive in 3-dimensional space. Even in an infinite string of stacked tetrahedra, no two tetrahedra will have the same orientation, because the helical pitch per cell is not a rational fraction of the circle. However, modified forms of this helix have been found which are rotationally repetitive, and in 4-dimensional space this helix repeats in rings of exactly 30 tetrahedral cells that tessellate the 3-sphere surface of the 600-cell, one of the six regular convex polychora.

A system of skew coordinates is a curvilinear coordinate system where the coordinate surfaces are not orthogonal, in contrast to orthogonal coordinates.

Generalized helicoid

In geometry, a generalized helicoid is a surface in Euclidean space generated by rotating and simultaneously displacing a curve, the profile curve, along a line, its axis. Any point of the given curve is the starting point of a circular helix. If the profile curve is contained in a plane through the axis, it is called the meridian of the generalized helicoid. Simple examples of generalized helicoids are the helicoids. The meridian of a helicoid is a line which intersects the axis orthogonally.

A proper reference frame in the theory of relativity is a particular form of accelerated reference frame, that is, a reference frame in which an accelerated observer can be considered as being at rest. It can describe phenomena in curved spacetime, as well as in "flat" Minkowski spacetime in which the spacetime curvature caused by the energy-momentum tensor can be disregarded. Since this article considers only flat spacetime—and uses the definition that special relativity is the theory of flat spacetime while general relativity is a theory of gravitation in terms of curved spacetime—it is consequently concerned with accelerated frames in special relativity.

References

  1. ἕλιξ Archived 2012-10-16 at the Wayback Machine , Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  2. Weisstein, Eric W. "Helicoid". MathWorld .
  3. "Double Helix Archived 2008-04-30 at the Wayback Machine " by Sándor Kabai, Wolfram Demonstrations Project.
  4. O'Neill, B. Elementary Differential Geometry, 1961 pg 72
  5. O'Neill, B. Elementary Differential Geometry, 1961 pg 74
  6. 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.
  7. 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 .
  8. Weisstein, Eric W. "Helix". MathWorld .

<https://www.merriam-webster.com/dictionary/geometrical%20pitch