Calculations at a logarithmic spiral. With this form of spirals, the radius increases proportionally with the spiral length. So the distance between two turnings increases exponentially with each turning for the factor f. The shape parameter determines the spiral's shape, this value mostly if far less than 1. It is easier to use the growth factor instead. The distance from the spiral start to the origin matches the size parameter a. Enter radius, number of revolutions or angle and shape parameter or growth factor. Choose the number of decimal places, then click Calculate. Please enter angles in degrees, here you can convert angle units.
Formulas:
r = a * ek*φ
n = φ / 360°
f = ( e2π )k
l = ( r - a ) / sin( arctan(k) )
p = l + r for n≤1
p = r - a * ek*(φ-360°) + l - ( a * ek*(φ-360°) - a ) / sin( arctan(k) ) for n>1
d = r + a * ek*(φ-180°) for φ>180°
h = a * ( ek*(φ-90°) + ek*(φ-270°) ) for φ>270°
ρ = r / cos( arctan(k) )
A = ( r² - a² ) / (4k)
pi:
π = 3.141592653589793...
Radius, parameter a, length, perimeter and diameter have a one-dimensional unit (e.g. meter), the area has this unit squared (e.g. square meter). The number of turnings, the shape parameter and the growth factor are dimensionless.
The first illustration of a logarithmic spiral can be found in 1525 by Albrecht Dürer in a work on the construction of geometric objects. René Descartes provided its mathematical definition in 1638. The Swiss mathematician and physicist Jakob Bernoulli discussed it in detail. Logarithmic spirals can be found in nature in nautilus shells, spiral galaxies and hurricanes, among other things. The Fibonacci spiral or golden spiral is a logarithmic spiral with the golden ratio φ=1.61803398875 as the growth factor. This can be seen, for example, in the arrangement of the seeds of a sunflower. Another type of spiral in which the distances between the turns remain the same is the Archimedean spiral.