Helix

4 min readFeb 23, 2022

(A Summary of it’s Properties and Applications)

Is this all we are? A necklace of chemicals? Where, in the double helix, does the soul lie? — Tess Gerritsen

The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The discipline owes it’s name to it’s use of ideas and techniques from differential calculus, though the modern subject often uses algebraic and purely geometric techniques instead. Although basic definitions , notations, and analytic descriptions vary widely.

One of such components of immense importance in the field of Differential Geometry are Helices

Definition:

A Helix is a space curve which is traced on the surface of a cylinder and cuts the generators at an constant angle ⍺ (say). Thus tangent to a helix makes a constant angle ⍺ with a fixed direction, this fixed line (direction) is known as axis or generator of the cylinder.

An annular strip (the region between two concentric circles) can be cut and bent into a helical strake that follows approximately the contour of a cylinder. Techniques of differential geometry are employed to find the dimensions of the annular strip that will best match the required curvature of the strake.

Helix is also called the curve of constant slope. Now since we’re quite acquainted with the term “helix” we shall discuss a few important properties of Helices (Helix is also called Cylindrical Helix)

Important Properties of Helices :

· For all Helices curvature bears constant ratio with torsion

It is a very important property of helices. Let a be a constant vector parallel to the generators of the cylinder and “t” the unit tangent vector to the helix. t◦𝛼 = acos . 𝛼 is the constant angle as defined above and |a|= a .

Differentiating with s we get, k n ◦ a = 0 [since t’=kn]

But k ≠ o for helix (where k is the curvature)

n◦a = 0. It shows that the principal normal is everywhere perpendicular to a i.e generators. But principal normal is everywhere perpendicular to the rectifying plane(containing t and b) hence the generators must be parallel to the rectifying plane. Also since generators are inclined at a constant angle with b.

Now differentiating n◦a = 0

(𝜏b — kt). a = 0 [since a is a constant function]

𝜏b.a — kt. a = 0

a𝜏sin 𝛼 — a k cos 𝛼=0 [since t.a=a cos 𝛼 and b.a=a sin 𝛼]

k/𝜏= tan 𝛼 = constant (𝜏 being the torsion)

Conversely, if k/𝜏 = constant, to show that the curve is a helix.

Let k/𝜏=c or k=c 𝜏, where c is constant.

We know t’ =k n = c 𝜏 n

b’= — 𝜏n,

t’=cb’=0

d/ds(t +cb)=0, which on integration gives

t+cb =a (constant vector)

taking the scalar product on each side with t,

1 = t.a

i.e t ◦ a=constant. Showing that t makes a constant angle 𝛼 with the fixed direction a, the curve is helix.

· If a curve is drawn on any cylinder and makes a constant angle with the generators

A helix is described on the surface of the circular cylinder or right circular helix. The axis of the helix coincides with the axis of the cylinder

The axis of the toroidal waveguide is a circular helix. The transformation of the orthogonal system ( X ̄ , Y ̄ ,Z ̄ ) from point A to the orthogonal system ( X , Y , Z ) at point K

A plane curve is a circle iff the corresponding helices are circular helices. A circular helix has constant band curvature and constant torsion. A curve is called a general helix or cylindrical helix if it’s tangent makes a constant angle with a fixed line in space,

· A spherical curve (A curve on the surface of a sphere, examples include the baseball cover, Seiffert’s spherical spiral, spherical helix) is a circle iff the corresponding Bertrand curves are circular helices.

Bertrand curves are the 3D curves, the curvature and torsion of which are linked by an affine non-linear relation(the linear case yields the helices). A curve is a Bertrand curve iff there exists a curve different from with the same principal normal line as.

fig1 and 2 are pictures of a cylindrical helix , for better understanding

Applications:

· Differential Geometry is not only the standard language for the formulaton of General Relativity, but it has found applications also in medical imaging, computer vision, Hamiltonian Mechanics , geometric design and information geometry.

The study helices facilitates us in better understanding of curves, manifolds(the higher-dimensional analogs of surfaces) and various complex structures

· Helical projections and its application in Screw modelling.

Helix play an important part in Biology as the DNA molecule is formed as two intertwined helices and many proteins have helical substructures, known as alpha helices.