Fisheye Types

Various types of fisheye lenses are available.
Here a short overview:
[table class=”table table-striped”]Type,gnomonic,stereographic,F-Theta,equal area,orthographic
lens class,wide angle,Fisheye,Fisheye,Fisheye,Fisheye
mapping function, r(\theta) = f \cdot tan \theta,r(\theta) = 2 \cdot f \cdot tan \frac{\theta}{2},r(\theta)=f \theta,r(\theta) = 2 \cdot f \cdot sin \frac{\theta}{2},r(\theta) = f \cdot sin \theta
normalized mapping function h(\theta)=\frac{r(\theta)}{f}, h(\theta) = tan \theta,h(\theta) = 2 \cdot tan \frac{\theta}{2},h(\theta)=\theta,h(\theta) = 2 \cdot  sin \frac{\theta}{2},h(\theta) = sin \theta
meridional scaling S_{m}={\frac {\mathrm {d} \mathbf {h} (\theta )}{\mathrm {d} \theta }},S_{m}={\frac {1}{\cos ^{2}\theta }},S_{m}={\frac {1}{\cos ^{2}{\frac {\theta }{2}}}},S_{m}=1,S_{m}=\cos {\frac {\theta }{2}},S_{m}=\cos \theta
sagittal scaling S_{s}={\frac {\mathbf {h} (\theta )}{\sin \theta }},S_{s}={\frac {1}{\cos \theta }},S_{s}={\frac {1}{\cos ^{2}{\frac {\theta }{2}}}},S_{s}={\frac {\theta }{\sin \theta }},S_{s}={\frac {1}{\cos {\frac {\theta }{2}}}},S_{s}=1
effective scaling S=\sqrt{S_{m} \cdot S_{s}},S={\frac {1}{\sqrt {\cos ^{3}\theta }}},S={\frac  1{\cos ^{2}{\frac  \theta 2}}},S={\sqrt {\frac {\theta }{\sin \theta }}},S=1,S={\sqrt {\cos \theta }}
N from S_{m}=S_{s}^{N},2,1,0,-1,\infty
Balance Deform. vs. Scaling B={\frac {2\,(N-1)}{N+1}},\frac{2}{3},0,-2,\infty,2
Curvature C={\frac {N-2}{3-N}},0,-{\tfrac  12},-{\tfrac  23},-{\tfrac  34},-1
Deformation D={\frac {S_{m}}{S_{s}}},D={\frac  1{\cos \theta }},D=1,D={\frac  {\sin \theta }\theta },D=\cos ^{2}{\frac  \theta 2},D=\cos \theta
maintains,-,angles,angular distances,areas,planar illuminance
AOV \alpha_{max},<180°,<360°,>= 360°,360°,180°
\alpha_{weak},54°,94°,115°,94°,66°
\alpha_{medium},75°,131°,159°,131°,90°
\alpha_{strong},102°,180°,217°,180°,120°
[/table]
Source of the various angles and formulae: wikipedia

Here the angle  \theta is measured in Radians !

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equal area lenses are also called “equisolid angle” or “flächentreu”
F-Theta lenses are also called “equidistant“, “linear scaled“, “äquidistant” or “angular
sterographic lenses are also called “ conform“or “ winkeltreu
orthographic lenses are also called” hemispherical“or “orthographisch

For details see:
equal area
F-Theta
gnomonic
orthographic
stereographic