Fisheye Types | OptoWiki Knowledge Base

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 !

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