A helix
(pl: helixes
or helices
) is a special kind of space curve, i.e. a smooth curve in three-space. As a mental image of a helix one may take the spring (although the spring is not a curve, and so is technically not a helix, it does give a convenient mental picture). A helix is characterised by the fact that the tangent line at any point makes a constant angle with a fixed line. A filled in
helix, for example a spiral staircase, is called a helicoid [1]. 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 ????
.
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HELIX TICKETS
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Types
Helices can be either right-handed or left-handed. With the line of sight being the helical axis, if clockwise movement of the helix corresponds to axial movement away from the observer, then it is called a right-handed helix. If anti-clockwise movement corresponds to axial movement away from the observer, 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 or flipped to look like a left-handed one unless it is viewed through a mirror, and vice versa.
Most hardware screws 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.
A
double helix typically consists geometrically of two congruent helices with the same axis, differing by a translation along the axis, which may or may not be half-way.
[2]
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 of a helix would be the
Corkscrew roller coaster at Cedar Point amusement park.
A
circular helix
has constant band
curvature and constant
torsion. The
pitch
of a helix is the width of one complete helix turn, measured along the helix axis.
A curve is called a
general helix
or
cylindrical helix
[3] 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
[4].
Mathematics
In
mathematics, a helix is a
curve in 3-
dimensional space. The following
parametrisation in
Cartesian coordinates defines a helix
[5]:
As the
parameter t
increases, the point (
x(t)
,
y(t)
,
z(t)
) traces a right-handed helix of pitch 2
p about the
z
-axis, in a right-handed coordinate system.
In
cylindrical coordinates (
r
, ?,
h
), the same helix is parametrised by:
The above example is an example of circular helix of radius 1 and pitch 2
p
.
Circular helix of radius
a
and pitch 2
pb
is described by the following parametrisation:
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
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 of the x, y or z components.
The length of a circular helix of radius
a
and pitch 2
pb
expressed in rectangular coordinates as
equals
, its Curvature#One dimension in three dimensions: Curvature of space curves|curvature is