Pitch angle of a spiral

In the geometry of spirals, the pitch angle^[1] or pitch^[2] of a spiral is the angle made by the spiral with a circle through one of its points, centered at the center of the spiral. Equivalently, it is the complementary angle to the angle made by the vector from the origin to a point on the spiral, with the tangent vector of the spiral at the same point.^[1] Pitch angles are frequently used in astronomy to characterize the shape of spiral galaxies.^[3]

The pitch angle α of a spiral

Logarithmic spirals are characterized by the property that the pitch angle remains invariant for all points of the spiral. Two logarithmic spirals are congruent when they have the same pitch angle, but otherwise are not congruent. For instance, only the golden spiral has pitch angle $${\displaystyle \arctan \left({\frac {\ln \varphi }{\pi /2}}\right)\approx 17^{\circ },}$$ where ${\displaystyle \varphi }$ denotes the golden ratio; logarithmic spirals with other angles are not golden spirals.^[1]

Spirals that are not logarithmic have pitch angles that vary by distance from the center of the spiral. For an Archimedean spiral the angle decreases with the distance, while for a hyperbolic spiral the angle increases with the distance.^[3]

References

1. Diedrichs, Danilo R. (February 2019), "Archimedean, Logarithmic and Euler spirals – intriguing and ubiquitous patterns in nature", The Mathematical Gazette, 103 (556): 52–64, doi:10.1017/mag.2019.7
2. Polezhaev, Andrey (2019), "Spirals, their types and peculiarities", in Tsuji, Kinko; Müller, Stefan C. (eds.), Spirals and Vortices: In Culture, Nature, and Science, The Frontiers Collection, Springer International Publishing, pp. 91–112, doi:10.1007/978-3-030-05798-5_4, ISBN 9783030057985
3. Savchenko, S. S.; Reshetnikov, V. P. (September 2013), "Pitch angle variations in spiral galaxies", Monthly Notices of the Royal Astronomical Society, 436 (2): 1074–1083, arXiv:1309.4308, doi:10.1093/mnras/stt1627