As a continuation from my previous post on 2D spirals, I decided to discuss 3-dimensional spirals. With 2-dimensional spirals there are only 2 variables (r, for radius, and θ), whereas there is a third variable in the description of 3D spirals (h for height). Thus, all 2D spirals can be extended to the third dimension by adding this third variable in the z-axis. Below I will detail 3 different types of 3D spiral.
A helix can be seen as a type of spiral as it is a curve in 3 dimensional space. There are many different types of helices, for example a conic helix which can be described as a 3D spiral on a conic surface.
Helices can be either right-handed or left-handed, meaning that helices form ‘enantiomers’.
Although there are many different formulae that produce a type of helix, the simplest parametric equations to produce one are:
A vortex is a phenomenon in fluid dynamics where the flow of the fluid “is rotating around an axis line, which may be straight or curved“.
The shape produced by a vortex can be described as a spiral due to its curved shape.
A Rhumb Line, or a loxodrome, is “is an arc crossing all meridians of longitude at the same angle, i.e. a path with constant bearing as measured relative to true or magnetic north.”
A rhumb line has an infinite number of revolutions as the separation between the lines decreases as it approaches the north or south pole (i.e. radius decreases); a rhumb line always spirals toward one of the pole. This is unique from the Archimedean spiral where the separation between the lines remains constant.