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).
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:
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 is a parabolic spiral that obeys the following polar equation:
It is a type of Archimedean 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.
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.
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:
In the centre, it is at an infinite distance from the pole; for θ starting at 0, r starts at infinity.
The lituus is a spiral with a polar equation:
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.
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
hence the name.
Special cases of this spiral include the golden spiral and the Fibonacci spiral, which approximates the golden spiral.
Let me know if you want to see a post on 3D spirals. M x