A helix () is a shape like a
corkscrew or
spiral staircase. It is a type of
smooth space curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
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".
A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called ''
helicoid''.
Properties and types
The ''pitch'' of a helix is the height of one complete helix
turn
Turn may refer to:
Arts and entertainment
Dance and sports
* Turn (dance and gymnastics), rotation of the body
* Turn (swimming), reversing direction at the end of a pool
* Turn (professional wrestling), a transition between face and heel
* 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
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.
For curves, the canonic ...
and constant
torsion.
A ''
conic helix
In mathematics, a conical spiral, also known as a conical helix, is a space curve on a right circular cone, whose floor plan is a plane spiral. If the floor plan is a logarithmic spiral, it is called '' conchospiral'' (from conch).
Parametr ...
'', also known as a ''conic spiral'', may be defined as a
spiral
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
Helices
Two major definitions of "spiral" in the American Heritage Dictionary are:[curvature
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.
For curves, the canonic ...]
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
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
Helices
Two major definitions of "spiral" in the American Heritage Dictionary are:[chirality
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from ...]
) 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
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a helix is a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
in 3-
dimensional space. The following
parametrisation in
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
defines a particular helix;
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 2''π'' (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 ''a''/''b'' (or pitch 2''πb'') is described by the following parametrisation:
:
:
:
Another way of mathematically constructing a helix is to plot the complex-valued function ''e
xi'' 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
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s,
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
arc length of a circular helix of radius ''a'' and slope ''a''/''b'' (or pitch = 2''πb'') expressed in rectangular coordinates as
: