Helix

Right helix (cos t , sin t , t ) with t = 0 to 4π

Crystalline right-handed molecular structure, based on the work of Lehn employees, from Helv. Chim. Acta., 2003, 86, 1598-1624

Natural twisting of the antennae of the grape (left spiral screw)

A helix is a curve in three-dimensional space located on a round cylinder or round cone , intersecting the generators at the same angle ^[1] .

A cylindrical helix is defined in rectangular coordinates by parametric equations of the form:

t\mapsto (a\cdot \cos t,a\cdot \sin t,b\cdot t)

{\ displaystyle t \ mapsto (a \ cdot \ cos t, a \ cdot \ sin t, b \ cdot t)}

{\ displaystyle t \ mapsto (a \ cdot \ cos t, a \ cdot \ sin t, b \ cdot t)}

,

or in another entry:

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

{\ displaystyle x (t) = a \ cdot \ cos t,}

{\ displaystyle x (t) = a \ cdot \ cos t,}

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

{\ displaystyle y (t) = a \ cdot \ sin t,}

{\ displaystyle y (t) = a \ cdot \ sin t,}

z(t)=b\cdot t

{\ displaystyle z (t) = b \ cdot t}

{\ displaystyle z (t) = b \ cdot t}

,

Where $a, b$ ${\ displaystyle a, b}$ $a, b$ Are real constants that are not equal to zero.

The projection of a cylindrical helix on a plane $x, y$ ${\ displaystyle x, y}$ $x, y$ is a circle .

A conical helix (also a helical helix ^[2] ) is determined by parametric equations of the form:

t\mapsto (a\cdot t\cdot \cos t,a\cdot t\cdot \sin t,b\cdot t)

{\ displaystyle t \ mapsto (a \ cdot t \ cdot \ cos t, a \ cdot t \ cdot \ sin t, b \ cdot t)}

{\ displaystyle t \ mapsto (a \ cdot t \ cdot \ cos t, a \ cdot t \ cdot \ sin t, b \ cdot t)}

,

or:

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

{\ displaystyle x (t) = a \ cdot t \ cdot \ cos t}

{\ displaystyle x (t) = a \ cdot t \ cdot \ cos t}

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

{\ displaystyle y (t) = a \ cdot t \ cdot \ sin t}

{\ displaystyle y (t) = a \ cdot t \ cdot \ sin t}

z(t)=b\cdot t

{\ displaystyle z (t) = b \ cdot t}

{\ displaystyle z (t) = b \ cdot t}

.

Projection of a helical line onto a plane $x, y$ ${\ displaystyle x, y}$ $x, y$ - the spiral of Archimedes .

A body shaped like a helix in colloquial speech is often called a spiral, which is not entirely correct, since in mathematics spirals are called a certain class of plane curves .

Content

1 Right and Left Helix Lines
2 Elements and properties
3 Examples of bodies in the form of a helix
4 notes

Right and Left Helix

There are mirror-symmetrical helix lines. The "right" helical lines are the lines generated by the " rule of the gimlet " or by the "rule of the right hand." This property of helix lines is called chirality - “right chirality” and “left chirality”. A pair of mirror-symmetrical helix lines are called enantiomorphs. If the coefficient $b$ ${\ displaystyle b}$ in the parametric task of a cylindrical helix in the right three coordinates is positive, then such a line is called "right", if negative - then "left".

The vast majority of threads used in mechanical engineering for fastening hardware have a “right” thread or a “right” chirality, that is, screwing is done clockwise. Left-handed threads are used very rarely in special applications, for example, to prevent self-unscrewing of pulleys from mechanism shafts.

Elements and Properties

Value $2\cdot \pi \cdot b$ ${\ displaystyle 2 \ cdot \ pi \ cdot b}$ called the pitch of the helix , geometrically this is the distance between adjacent turns of the line, counted along the generatrix of the cylinder .

All helical lines are slope lines , that is, tangents to them form a constant angle with some constant direction. As with any slope, the curvature of a cylindrical helix $C$ ${\ displaystyle C}$ and torsion $S$ ${\ displaystyle S}$ constant at any point and equal to:

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

{\ displaystyle C = {\ frac {| a |} {a ^ {2} + b ^ {2}}}}

,

S={\frac {b}{a^{2}+b^{2}}}

{\ displaystyle S = {\ frac {b} {a ^ {2} + b ^ {2}}}}

.

Length element $d L$ ${\ displaystyle dL}$ :

dL=dt\cdot {\sqrt {a^{2}+b^{2}}}

{\ displaystyle dL = dt \ cdot {\ sqrt {a ^ {2} + b ^ {2}}}}

.

Angle $\Phi$ ${\ displaystyle \ Phi}$ between the tangent to the cylindrical helix and the tangent to the circumference of the cylinder at the same point is called the helical angle , it is equal to:

\Phi =\arctan({\tfrac {b}{a}})

{\ displaystyle \ Phi = \ arctan ({\ tfrac {b} {a}})}

.

Helix examples of bodies

The following molecules , for example, have the shape of a helix or helicoid :

DNA is a double helix (double screw),
RNA
actin filament - double helix,
calmodulin
molecules having chirality
asparaginase .

Many parts of machines and mechanisms also have forms of helical lines - springs , part of screw drills , connecting screws , bolts , studs , screws ( screws ) of meat grinders , extruders , Archimedes screw , screws of snow blowers and others (they realize a helical surface - a helicoid ).

Right screw spring

Notes

↑ Helix - an article from the Mathematical Encyclopedia . E.V. Shikin
↑ Helix - an article from the Great Soviet Encyclopedia . E. G. Poznyak.

[1] Helix - an article from the Mathematical Encyclopedia . E.V. Shikin

[2] Helix - an article from the Great Soviet Encyclopedia . E. G. Poznyak.