The equations of the basic spirals are given in polar coordinates. The is the radius and is the angle measured from the positive horizontal axis.The and are constants.
Archimedes Spiral
http://mathworld.wolfram.com/ArchimedesSpiral.html
Fermat Spiral
http://mathworld.wolfram.com/FermatsSpiral.html
Logarithmic Spiral
http://mathworld.wolfram.com/LogarithmicSpiral.html
Golden Spiral
The equation in polar coordinates is the same as the logarithmic spiral but the the constant has a special value
The constant is in units of radians. is the golden number.
http://mathworld.wolfram.com/GoldenSpiral.html
Hyperbolic Spiral
http://mathworld.wolfram.com/HyperbolicSpiral.html
Lituus Spiral
http://mathworld.wolfram.com/Lituus.html
Cornu Spiral
The Cornu spiral is also known as the clothoid or Euler’s spiral. The mathematical equation is not as simple as the others shown above. Please refer to the link below.
http://mathworld.wolfram.com/CornuSpiral.html
Circle Involute
http://mathworld.wolfram.com/CircleInvolute.html
http://mathworld.wolfram.com/Involute.html
Cotes Spiral
A spiral that gives the solution to the central orbit problem under a radial force law. Please refer to the link below.
http://mathworld.wolfram.com/CotesSpiral.html
Nielsen Spiral
http://mathworld.wolfram.com/NielsensSpiral.html
Polygonal Spiral