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Field of View

(usually horizontal) extend of an object that’s visible on a sensor.

Sometimes the HFOV (horizontal field of view), sometimes the VFOV (Vertical Field of View) and sometimes the DFOV (Diagonal Field of View) matters. It’s of utmost importance to state clearly which matters.

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
meridionale Skalierung 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
sagittale Skalierung 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
maintaines - 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