# Field of View

From PanoTools.org Wiki

The angle of view of a photograph or camera is a measure of the proportion of a scene included in the image. Simply said: How many degrees of view are included in an image. A typical fixed lens camera might have an angle of view of 50°, a fisheye lens can have an angle of view greater than 180° and a full equirectangular or cylindrical panorama would have an angle of view of 360°.

Most people speak of field of view when in fact they mean angle of view. Field of view is the distance covered by a projection at a certain distance. So if an image exactly shows a 2 meter wide object at 1 meter distance, then the field of view is 2 meter (and the angle of view is 90°). Angle of view is also known as angle of coverage.

From here on and on the rest of the wiki we will only speak of field of view (although we should speak of angle of view).

Field of view is often abbreviated as FoV. Usually field of view refers to the horizontal field of view (hFoV) of an image. Some applications make use of the vertical field of view (vFoV) which can be calculated from the Aspect Ratio of the image:

For rectilinear images:

A s p e c t R a t i o = t a n ( h F o V 2 ) t a n ( v F o V 2 ) {\displaystyle AspectRatio={\frac {tan({\frac {hFoV}{2}})}{tan({\frac {vFoV}{2}})}}}

For fisheye images (approximation):

A s p e c t R a t i o = h F o V v F o V {\displaystyle AspectRatio={\frac {hFoV}{vFoV}}}

## Conversion from focal length

The other standard measure of the width or narrowness of a lens is Focal Length.

Assuming a rectilinear lens, the field of view can be calculated like this ( s i z e {\displaystyle size} being either width or height for the respective FoV):

F o V = 2 ∗ a t a n ( s i z e 2 ∗ F o c a l L e n g t h ) {\displaystyle FoV=2*atan\left({\frac {size}{2*FocalLength}}\right)}

Please note that this is an approximation. The exact values depend on the location of the entrance pupil. More information on that in Rik Littlefield's paper. See Fisheye Projection for formulas for Fisheyes[*].

Please also note that some software may modify somewhat the above relationship between FoV and focal length. For example, Hugin takes into account ratio of input image (see beginning of function SrcPanoImage::calcFocalLength() for details).

## Conversion from horizontal to vertical and vice versa

For fisheye (approximation) and equirectangular images:

v F o V = h F o V ∗ h e i g h t w i d t h {\displaystyle vFoV=hFoV*{\frac {height}{width}}\ }

h F o V = v F o V ∗ w i d t h h e i g h t {\displaystyle hFoV=vFoV*{\frac {width}{height}}\ }

For rectilinear images:

v F o V = 2 ∗ a t a n ( t a n ( h F o V 2 ) ∗ h e i g h t w i d t h ) {\displaystyle vFoV=2*atan\left(tan\left({\frac {hFoV}{2}}\right)*{\frac {height}{width}}\right)}

h F o V = 2 ∗ a t a n ( t a n ( v F o V 2 ) ∗ w i d t h h e i g h t ) {\displaystyle hFoV=2*atan\left(tan\left({\frac {vFoV}{2}}\right)*{\frac {width}{height}}\right)}

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