How to Find the Area and Arc Length of a Logarithmic Spiral
A logarithmic spiral, also called an equiangular spiral, is one whose distance from the center increases exponentially with every turn. The most convenient way to describe a logarithmic spiral mathematically is with a polar coordinate equation
r(θ) = a*b^θ
where θ is the angle (in radians) of a point with respect to the positive x-axis, and r is the distance of the point to the origin. The constants a and b determine how fast the spiral grows. Logarithmic spirals occur naturally in the curl of snail shells and in the spiral patterns of pine cones and sunflower heads, just to name a few examples. Logarithmic spirals are a salient feature of many fractal designs as well.
A logarithmic spiral is distinct from an Archimedes spiral, whose equation is r(θ) = a+b*θ and whose distance from the center grows at a constant fixed rate of b with every turn. Another property of logarithmic spirals is that they have no beginning. If you exampine negative values of θ in the equation r(θ) = a*b^θ, you will see that the spiral becomes smaller and smaller. However, if you expand the range of an Archimedes spiral equation to include negative values of θ, the spiral doubles back on itself.
You can use calculus to find the area enclosed by an logarithmic spiral as well as the arc length of a segment of the spiral's curve
Logarithmic SpiralsClick thumbnail to view full-size
Area Enclosed by a Logarithmic Spiral
To find the area enclosed by a logarithmic spiral, you need to compute a definite integral whose limits are consecutive turns, that is, the limits of the definite integral must be N to N+2π, where N is any real number.
Though it may seem more natural to use the limits negative infinity to N, you will end up counting some regions multiple times, and some regions an infinite number of times if you use these limits. To avoid over-counting, use the limits N to N+2π, where N+2π is the maximum number of turns
For example, let's compute the total area enclosed by the logarithmic spiral r(θ) = e^(θ/3) up to θ = 4π. This means our limits of integration are θ = 2π to θ = 4π. Using the area integral formula for polar coordinates, we have
= (1/2) ∫ [e^(θ/3)]^2 dθ, over [2π, 4π]
= (3/2)e^(θ/3), over [2π, 4π]
= (3/2)e^(4π/3) - (3/2)e^(2π/3)
Arc Length of a Logarithmic Spiral
To compute the arc length of a logarithmic spiral, we need to use the arc length integral formula for polar coordinates. As an example, let's find the length of curve of the spiral r(θ) = e^(θ/3) from between θ = 2π and θ = 4π. This gives us
Arc Length Integral
= ∫ sqrt[ r(θ)^2 + r'(θ)^2 ] dθ
= ∫ sqrt[ e^(2θ/3) + (1/9)e^(2θ/3) ] dθ
= [sqrt(10)/3] * ∫ e^(θ/3) dθ
= sqrt(10)*e^(θ/3) + C
Evaluating this between the limits θ = 2π and θ = 4π gives us
= sqrt(10)*e^(4π/3) - sqrt(10)*e^(2π/3)
What if we change the lower limit to negative infinity? That is, compute the total length of the spiral up to a given index. To do this for the example above, we replace 2π with -∞, which gives us
Total Arc Length Up To 4π
= sqrt(10)e^(4π/3) - sqrt(10)*e^(-∞/3)
= sqrt(10)e^(4π/3) - sqrt(10)*0
Thus, even though the logarithmic spiral has no beginning, its arc length from the "beginning" to any arbitrary end is a finite amount.
Approximate Construction of a Logarithmic Spiral
Drawing a true logarithmic spiral is difficult without the aid of a computer graphics program, but you can make a reasonable approximation of a spiral by piecing together circular arcs or circular sectors of constant angle whose radii increase by constant factor. For example, if you use quarter-circular arcs whose radii increase by a factor of (sqrt(5) + 1)/2 ≈ 1.618, aka the golden mean, you can construct an approximate Fibonacci spiral. This is shown in the image above.
The arc length and area of an approximation spiral can be found by computing the sum of a geometric series. To do these calculations, let's say the angle of the circular arcs is A radians, and the constant factor of radius increase is F. If the largest of the circular arcs has a radius of X, then the total arc length of the approximation spiral is given by the sum
Arc Length = XA + XA/F + XA/F^2 + XA/F^3 + ...
= XA/(1 - 1/F)
= XAF/(F - 1)
For example, suppose an approximated Fibonacci spiral has A = π/2 radians (90 degrees, quarter circle), F = (sqrt(5)+1)/2, and X = 10. Then the total length of the spiral is
10*(π/2)*[(sqrt(5)+1)/2] / [(sqrt(5)-1)/2] = 41.124.
To find the total area of all the circular sectors, the geometric series is
Area = (AX^2)/2 + (AX^2)/(2F) + (AX^2)/(2F^2) + (AX^2)/(2F^3) + ...
= [(AX^2)/2] / (1 - 1/F)
= AFX^2 / (2F - 2)
Using the same example, the total area of the circular sectors of the Fibonacci spiral approximation is
(π/2)*[(sqrt(5)+1)/2]*10^2 / (sqrt(5) - 1) = 205.620.