Archimedian Spirals Constructed on Polargraphic Paper with Geometric Algebra
Single Turn Spiral
This is the trajectory of a point that travels .27... rads (unit of distance from center) every 10° of rotation from 0 to 360° , going 10 rads in the process from the center to the outside, making one turn of the spiral for the 360° rotation.
This constant increase in the distance from the center, lets call it Delta rad or D(r), here is .027.../degree, or .27... rads/ 10° counter clockwise rotation we used in this construction. Such a constant increase defines an Archimedian spiral, after the Greek mathematician, Archimedes, who studied them extensively.
Thus, the total rads, R(n), at any step n, equals the previous total, R(n-1) plus D(r), or
R(n) = R(n-1) + D(r) = n * D(r)
In this case of the one turn spiral,
R(n) = 36 * .277 where n=36, the total number of 10° steps taken from the initial position at 0°. Thus after the 36 steps to complete the circle, 36 * .27...rads = 10 rads.
Created: 11/24/97Updated: 12/7/97