Spirals: 3D

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.

Helix

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’.

Two_Types_of_Helix.svg.png

Although there are many different formulae that produce a type of helix, the simplest parametric equations to produce one are:

x(t) = \cos(t),\,y(t) = \sin(t),\,z(t) = t.\,

Rising_circular.gif

Source: Wikipedia

Vortex

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.

vortex_schematic.gif

Rhumb Line

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.”

Source: Wikipedia

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.

KUGSPI-9_Loxodrome.gif

Rhumb Line

M x

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2 comments

  1. Interesting to note that vortexes are naturally formed when water flows out into a pipe, for example a tap in a sink. The direction of rotation is then determined by the rotation of the planet earth. In the North hemisphere anticlockwise and in the south hemisphere clockwise. This is the so-called Coriolis effect. However, this is a small effect, so it can be countered if the water released from tap is forced in the opposite direction, the shape of the sink and plughole, friction, and other factors.

    Liked by 2 people

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