Fisheye Types | OptoWiki Knowledge Base

Various types of fisheye lenses are available.
Here a short overview:

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°

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