2 A B C D E F G H I K L M N O P R S T U V W

Fisheye Types

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 !

image/svg+xml3/2 πP(r,φ,θ)0rθr1/2 ππ0φxyz

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