Logarithmic Spiral Equiangular Geometry

Worked Example

Let's consider a logarithmic spiral with an initial radius a = 1 and a growth parameter b = 0.5. We want to determine its geometric properties when evaluated at an angle θ = 1 radian.

The calculation is as follows:

1. Position Vector:
   Radius r = a * e^(b * θ)
   r = 1 * e^(0.5 * 1) = e^0.5 ≈ 1.6487212707001282
   x = r * cos(θ) = 1.6487212707001282 * cos(1) ≈ 0.8908079042931287
   y = r * sin(θ) = 1.6487212707001282 * sin(1) ≈ 1.3873511113297634

2. Radial Tangent Angle (α):
   α = atan2(1, b) = atan2(1, 0.5) ≈ 1.1071487177940904 radians

3. Arc Length from zero to θ:
   s = a * √(1 + b²) * (e^(b*θ) - 1) / b
   s = 1 * √(1 + 0.25) * (e^0.5 - 1) / 0.5
   s = √(1.25) * 2 * (1.6487212707001282 - 1) ≈ 1.4505848597355293

4. Curvature (κ):
   κ = 1 / (r * √(1 + b²))
   κ = 1 / (1.6487212707001282 * √(1.25)) ≈ 0.5424975142220966

Experiment

You can investigate the equiangular property of the logarithmic spiral using the Logarithmic Spiral Equiangular Geometry module.

1. Set the Initial radius to 1.
2. Set the Growth b to 0.5.
3. Set the Angle radians to 1 and then 2. Observe that while the curvature and position change, the Outward radial tangent angle remains exactly the same.
4. Try setting the Growth b to 0. The spiral degenerates into a circle, the tangent angle becomes exactly π/2, and the radius multiplier becomes 1.0.

Limitations

The continuous representation of the logarithmic spiral stretches to an infinite number of turns toward the origin as θ → -∞. In standard floating-point representation, values for highly negative θ combined with a positive growth factor b may underflow to absolute zero. Similarly, setting a large positive angle results in exponential growth that can overflow. The provided formulation assumes the origin is well-defined as the asymptotic limit but does not technically exist within the standard real parameterization interval unless analyzed carefully.

References

For a thorough mathematical treatment of logarithmic spirals and their properties, see the lecture notes at Logarithmic Spiral Notes.