Spirals are defined as: a curve that winds around a point while moving further from that point.
The three spirals that were discussed earlier are all different variations of this same concept.
Spiral
In the case of the Archimedian/Arithmetic Spiral the equation that was derived from Archimedes' research is as follows:
$$r = a(θ)+b$$ where:
a = controls the distance between each loop
b = moves the center point of the spiral outward from the origin
Press the "Draw" button below and watch how the spiral is formed.
Think back to Archimedes using a compass with one end stuck at the origin, call this A, and the other a certain distance away, call this C.
Then imagine him turning the compass in a circular motion around the fixed point, the origin, while moving C at a constant rate away from A.
That distance from the first leg to the second is your r value and the angle away from the Polar Axis is the θ value.
One of the main characteristics of the Archimedian Spiral is that each rotation is the same distance from the previous one.
Hence when you draw a ray straight out from the origin each time it intersects a successive turn of the spiral, there will be a constant distance that separates each turn.
Drag the a and b sliders to see how they individually change how the spiral looks.
Spiral
From the Archimedean formula other mathematicians have derived other spirals.
These spirals, their formulas and a pitcure of the base spiral, meaning its centered at the origin, are provided in the table below (Weisstein).
Archimedian
$$r = a(\theta)^{\frac{1}{n}}+b$$
n = 1
Hyperbolic
$$r = \frac{a}{\theta}$$
n = -1
Fermats
$$r = a(\theta)^{\frac{1}{2}}$$
n = 2
Lituus
$$r = a(\theta)^{\frac{-1}{2}}$$
n = -2
In the video below, you have the opportunity to learn how to draw an archimedean spiral by hand.
Start the video from the beginning and stop it after 2 minutes 20 seconds and move on to the next section.
