Spirals: 2D

A spiral is a curve which “emanates from a point, moving farther away as it revolves around the point”.

There are many different types of spirals depending on the formulae that create them. In this blog post, I will discuss two-dimensional spirals (note that there also exist a variety of three dimensional spirals).

Archimedean Spiral

This is a spiral named after the famous Greek mathematician Archimedes, who was the first to describe it in his book On Spirals. It is described by the following polar equation:

\, r=a+b\theta

where a and b are real numbers.

By changing parameter a the spiral will turn and parameter b controls the distance between successive turnings.

Fermat’s Spiral

Fermat’s spiral is a parabolic spiral that obeys the following polar equation:

r=\pm \theta ^{1/2}\,

It is a type of Archimedean spiral.

Euler Spiral

Also known as a Cornu Spiral, it is a curve whose curvature grows as the distance from the origin increases; “the curvature of a circular curve is equal to the reciprocal of the radius”.

The parameter form consists of two equations with Fresnel’s intervals, which can only be solved approximately.

S(x)=\int_0^x \sin(t^2)\,\mathrm{d}t,\quad C(x)=\int_0^x \cos(t^2)\,\mathrm{d}t.

These integrals, and hence the Euler spiral, can be used to describe the energy distribution of Fresnel’s diffraction at a single slit in wave theory.

Hyperbolic Spiral

First conceived by Pierre Varignon in 1704, the hyperbolic spiral is a transcendental curve, meaning that “it is an analytic function that does not satisfy a polynomial equation“. It is the opposite of an Archimedean spiral and thus has the following polar equation:

r={\frac  {a}{\theta }}

In the centre, it is at an infinite distance from the pole; for θ starting at 0, r starts at infinity.

Lituus

The lituus is a spiral with a polar equation:

r^2\theta = k \,

where k is a constant. Hence, the angle is inversely proportional to the square of the radius. 

The curve was named after the ancient Roman lituus (a curved augural staff or war-trumpet) by Roger Cotes in a collection of papers published in 1722.

Logarithmic Spiral

The logarithmic spiral is a self-similar spiral, which often appears in nature. It was first described by Descartes, but was studied in depth by Jacob Bernoulli who called it “the marvellous spiral”. It has a polar equations of

 r = ae^{b\theta}\,

or \theta = \frac{1}{b} \ln(r/a),

hence the name.

Special cases of this spiral include the golden spiral and the Fibonacci spiral, which approximates the golden spiral.

Sources: 1 | 2 | 3

Let me know if you want to see a post on 3D spirals. M x

 

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