Golden Spiral - Formula

Formula

The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of the growth factor b:

or

with e being the base of natural logarithms, a being an arbitrary positive real constant, and b such that when θ is a right angle (a quarter turn in either direction):

Therefore, b is given by

The numerical value of b depends on whether the right angle is measured as 90 degrees or as radians; and since the angle can be in either direction, it is easiest to write the formula for the absolute value of (that is, b can also be the negative of this value):

for θ in degrees;

for θ in radians.

An alternate formula for a logarithmic and golden spiral is:

where the constant c is given by:

which for the golden spiral gives c values of:

if θ is measured in degrees, and

if θ is measured in radians.