アルキメデスの螺旋

アルキメデスの螺旋 ：Archimedean spiral †

アルキメデスの螺旋（らせん　Archimedes' spiral）は極座標の方程式r = a・θによって表される曲線である。等間隔の渦巻きである。 θが負の場合も含めると、y軸に対して線対称となる。

• The Archimedes' spiral (or spiral of Archimedes) is a kind of Archimedean spiral.

  Archimedes' spiral 
  r=a・θ　.

• Sometimes the term Archimedean spiral is used for the more general group of spirals
  
  The normal Archimedean spiral occurs when x = 1. Other spirals falling into this group include the hyperbolic spiral, Fermat's spiral, and the lituus.
• The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd century BC Greek mathematician Archimedes. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation

  r=a+b・θ　

  with real numbers a and b. Changing the parameter a will turn the spiral, while b controls the distance between successive turnings.

• This Archimedean spiral is distinguished from the logarithmic spiral by the fact that successive turnings of the spiral have a constant separation distance (equal to 2πb if θ is measured in radians), while in a logarithmic spiral these distances form a geometric progression.
  

フェルマーの螺旋　Fermat's spiral †

Fermat's spiral (also known as a parabolic spiral) follows the equation in polar coordinates (the more general Fermat's spiral follows r^ 2 = a^2・θ.) It is a type of Archimedean spiral.

Fermat's spiral traverses equal annuli in equal turns. The full model proposed by H Vogel in 1979 is

where θ is the angle, r is the radius or distance from the center, and n is the index number of the floret and c is a constant scaling factor. The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers.