In the geometry ofspirals, thepitch angle^[1] orpitch^[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 thecomplementary angle to the angle made by the vector from the origin to a point on the spiral, with thetangent vector of the spiral at the same point.^[1] Pitch angles are used to characterize the steepness of spirals, such as inastronomy to describespiral galaxies.^[3]

Logarithmic spirals, for example, 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 thegolden spiral has pitch angle$${\displaystyle \arctan \left(\frac{\ln \phi }{\pi /2}\right)\approx {17}^{\circ },}$$where${\displaystyle \phi }$ denotes thegolden 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 anArchimedean spiral the angle decreases with the distance, while for ahyperbolic spiral the angle increases with the distance.^[3] The equation for determining pitch angle is,^[4]

$${\displaystyle \psi =\arctan \left(\frac{r}{\frac{dr}{d\theta }}\right)-\frac{\pi }{2}}$$

• Spirals

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