Editing Gudermannian function (section)

==Applications==

[[File:Distance in the half-plane model 3.png|thumb|right|upright=1.5|Distance in the [[Poincaré half-plane model]] of the [[hyperbolic plane]] from the apex of a semicircle to another point on it is the inverse Gudermannian function of the central angle.]]

*The [[angle of parallelism]] function in [[hyperbolic geometry]] is the [[Angle#Combining angle pairs|complement]] of the gudermannian, <math>\mathit{\Pi}(\psi) = \tfrac12\pi - \operatorname{gd} \psi.</math>
* On a [[Mercator projection]] a line of constant latitude is parallel to the equator (on the projection) at a distance proportional to the anti-gudermannian of the latitude.
* The Gudermannian function (with a complex argument) may be used to define the [[transverse Mercator projection]].<ref>{{harvp|Osborne|2013}} p. 74</ref>
* The Gudermannian function appears in a non-periodic solution of the [[inverted pendulum]].<ref>{{harvp|Robertson|1997}}</ref>
* The Gudermannian function appears in a moving mirror solution of the dynamical [[Casimir effect]].<ref>{{harvp|Good|Anderson|Evans|2013}}</ref>
* If an infinite number of infinitely long, equidistant, parallel, coplanar, straight wires are kept at equal [[electric potential|potentials]] with alternating signs, the potential-flux distribution in a cross-sectional plane perpendicular to the wires is the complex Gudermannian function.<ref>{{harvp|Kennelly|1928}}</ref>
* The Gudermannian function is a [[sigmoid function]], and as such is sometimes used as an [[activation function]] in machine learning.
* The (scaled and shifted) Gudermannian function is the [[cumulative distribution function]] of the [[hyperbolic secant distribution]].
* A function based on the Gudermannian provides a good model for the shape of [[spiral galaxy]] arms.<ref>{{harvp|Ringermacher|Mead|2009}}</ref>