# Angle of view

Horizontal angle of view , vertical angle of view and diagonal angle of view${\ displaystyle \ alpha _ {h}}$${\ displaystyle \ alpha _ {v}}$${\ displaystyle \ alpha _ {d}}$

In photography, the angle of view is the angle in the subject space that is limited by the edges of the recording format. In the usual rectangular image format, the image angle usually means the value associated with the image diagonal . Horizontal and vertical image angles are smaller and are to be referred to in full length as horizontal image angle and vertical image angle , respectively. If the width and height are usually greater, the vertical image angle is the smallest and the diagonal the largest of the three angles.

Apart from the image format - height , width or diagonal of the image - the angle of view is essentially only determined by the current focal length of the camera lens . The formula below provides the diagonal angle of view when the lens is set to "infinite" : ${\ displaystyle h}$${\ displaystyle b}$${\ displaystyle d}$${\ displaystyle \ textstyle \ alpha}$ ${\ displaystyle \ textstyle f}$${\ displaystyle (\ infty)}$

${\ displaystyle \ alpha _ {d} = 2 \ cdot \ arctan \ left ({\ frac {d} {2 \ cdot f}} \ right)}$

In contrast, when depicting closer objects, the image distance is greater than , whereby the image angle is correspondingly reduced. ${\ displaystyle b}$${\ displaystyle f}$${\ displaystyle \ textstyle \ alpha}$

According to the formula, the angle of view can be a maximum of 180 ° , because the value range of the arctangent function is limited to −90 ° and + 90 °. Fisheye lenses still have an angle of view that extends beyond 180 °, but do not produce a collinear image, i.e. That is, straight lines are shown curved so that the formula for the angle of view cannot be used here. In the case of lenses with pronounced distortions , the local focal length of the lens corresponding to the recording format must be used at the edges or corners of the recording format. Otherwise, the formula applies without restriction to collinear imaging lenses with a focal length that is independent of the angle of view.

## Directional dependence

Except for a circular recording format, the size of the angle of view depends on the direction - in relation to the horizontal of the recording medium - in which it is determined.

The particular importance of the diagonal angle of view is that the diagonal of a rectangle corresponds to its greatest extent. Thus, for a given focal length and recording format, the diagonal is always the largest angle of view .

The direction of the respective angle of view relative to the horizontal is at

• horizontal angle of view: 0 ° or 180 °
• vertical angle of view: ± 90 °
• diagonal angle of view: ${\ displaystyle \ textstyle \ pm \ arctan \ left ({\ frac {\ text {height}} {\ text {width}}} \ right)}$

When comparing different recording formats, it should therefore be noted that the diagonal angle of view of the two formats can differ in their direction . A comparison based on the diagonal angle of view is only meaningful if the recording formats to be compared have approximately the same aspect ratios .

By cropping the exposed image or by restricting the recording format when recording by means of masks in front of the image plane , recording formats of any shape can be realized. In this case, when specifying the angle of view, one refers to an imaginary rectangle that just describes the actual recording format.

A special case is the circular recording format, here the angle of view is independent of the direction and constant. This also applies to the (circular) image circle .

Angle of view and field of view. For the sake of illustration, only the vertical angle of view is shown

## Field of view

As a field of view (engl. Field of view , FOV) is the one area in the object space referred to, which is spanned by the horizontal and vertical angle.

## Maximum possible angle of view

With a single converging lens or a pinhole , the maximum angle of view can theoretically be up to 180 °. In the case of lenses, however, the light has to pass through several openings one after the other, which act as apertures . This also includes the lenses and their mounts. Thus the usable angle of view is limited. Image errors can limit the actually useful angle of view even further. Special lens constructions such as B. fisheye lenses can still image angles of 180 ° - or even beyond. The maximum possible angle of view of a lens is specified in the technical data. This gives the diagonal of the largest usable recording format : ${\ displaystyle \ textstyle \ omega}$

${\ displaystyle d _ {\ mathrm {max}} = 2 \ cdot f \ cdot \ tan \ left ({\ frac {\ omega} {2}} \ right)}$

The image circle diameter corresponds to the diagonal of the largest usable recording format${\ displaystyle \ textstyle d _ {\ text {max}}}$ .

## Normal viewing angle and normal focal length of a photo lens

Values ​​between 40 ° and 55 ° are given for the viewing angle with a natural impression that one has of the scene . Viewing photos leads to the same impression if they were taken with the same angle of view. As a rule of thumb, the focal length of the lens should be approximately the same size as the diagonal of the image format ( ). With this assumption, the angle of view is about 53 °: ${\ displaystyle f \ approx d _ {\ mathrm {max}}}$

${\ displaystyle \ alpha = 2 \ cdot \ arctan \ left ({\ frac {1} {2}} \ right) = 53 {,} 13 ^ {\ circ}}$

The matching normal focal length for the 35 mm format of 36 mm × 24 mm would be: ${\ displaystyle f _ {\ mathrm {N}}}$

${\ displaystyle f _ {\ mathrm {N}} = d = \ mathrm {({\ sqrt {{36 ^ {2}} + {24 ^ {2}}}}) \, mm = 43 {,} 3 \ , \ mathrm {mm}}}$

However, the 35mm focal length has become commonplace , whereby the image angle falls into the above-mentioned range between 40 ° and 55 °: ${\ displaystyle f _ {\ mathrm {N}} = 50 \, \ mathrm {mm}}$${\ displaystyle \ alpha _ {d}}$

${\ displaystyle \ alpha _ {d} = 2 \ arctan {\ frac {d} {2f _ {\ mathrm {N}}}} = 2 \ arctan {\ frac {43 {,} 3} {2 \ times 50} } \ approx 46 {,} 8 ^ {\ circ}}$  diagonal angle of view

The other two angles of view are:

${\ displaystyle \ alpha _ {h} = 2 \ arctan {\ frac {h} {2f _ {\ mathrm {N}}}} = 2 \ arctan {\ frac {36} {2 \ times 50}} \ approx 39 {,} 6 ^ {\ circ}}$ horizontal angle of view,
${\ displaystyle \ alpha _ {v} = 2 \ arctan {\ frac {v} {2f _ {\ mathrm {N}}}} = 2 \ arctan {\ frac {24} {2 \ times 50}} \ approx 27 {,} 0 ^ {\ circ}}$ vertical angle of view.

If the aspect ratio is different from that of 35mm film, the proportions of the horizontal and vertical image angles are different. For this reason, the diagonal angle is usually used to compare different images, since this always relates to the maximum image circle regardless of the aspect ratio . With an image angle of 46.8 °, there is always an image diagonal for infinite object widths, regardless of the aspect ratio, which is around 13.5% smaller than the focal length.

A wide-angle lens has a significantly larger and a telephoto lens a significantly smaller angle of view than the normal angle of view.

The relative image scale relates the actual image scale of an optical image to the image scale at the normal focal length of the respective optical system.

## Viewing angle

Analogous to the angle of view, the angle of view on the object side is defined as ${\ displaystyle \ textstyle \ beta}$

${\ displaystyle \ beta = 2 \ cdot \ arctan \ left ({\ frac {l} {2 \ cdot g}} \ right)}$

where stands for the horizontal, vertical or diagonal extent of the object to be imaged and for the distance of the object (more precisely: for the object distance ). ${\ displaystyle \ textstyle l}$${\ displaystyle \ textstyle g}$

The viewing angle (also apparent size ) describes the angle at which an object appears at a given extent and distance .

One can visualize the viewing angle by reversing the beam path.

In order to be able to fully depict an object on the recording medium, the horizontal and vertical image angle must not be smaller than the corresponding viewing angle , so the following must apply:

${\ displaystyle {\ frac {d _ {\ text {v; h}}} {f}} \ geq {\ frac {l _ {\ text {v; h}}} {g}}}$

If a motif is to be shown sufficiently sharp at close range and with optics with a large opening angle , the object distance must also be included in the field of view (see section Macro Photography ). The effective angle of view must not be smaller than the corresponding viewing angle :

${\ displaystyle {\ frac {d _ {\ text {v; h}}} {f}} - {\ frac {d _ {\ text {v; h}}} {g}} \ geq {\ frac {l_ { \ text {v; h}}} {g}}}$

## Angle of view and viewing angle in photography

If you change the recording format and the focal length by the
same factor , the angle of view (here 69 °) does not change .${\ displaystyle \ textstyle d}$${\ displaystyle \ textstyle f}$ ${\ displaystyle \ textstyle \ alpha}$

Angle of view and viewing angle are among the most important photographic design tools, the former solely dependent on the focal length of the lens used and the recording format of the camera, while the latter solely depends on the size and distance of the object to be photographed.

Accordingly , if the photographer wants to change the angle of view , this can either be done by changing the focal length, e.g. B. do with the help of interchangeable lenses , converters or by using a zoom lens , but alternatively also by choosing a different recording, d. H. Sensor or recording format , subsequent trimming of the recording or the merging of several individual recordings ( stitching ) into an overall picture.

It applies: If you change the recording format and the focal length by the same so-called Format factor , the angle of view does not change . This relationship results directly from the definition of the angle of view: ${\ displaystyle \ textstyle d}$ ${\ displaystyle \ textstyle f}$ ${\ displaystyle \ textstyle x}$ ${\ displaystyle \ textstyle \ alpha}$

${\ displaystyle \ alpha = 2 \ cdot \ arctan \ left ({\ frac {d} {2 \ cdot f}} \ right) = 2 \ cdot \ arctan \ left ({\ frac {d \ cdot x} {2 \ cdot f \ cdot x}} \ right)}$

The viewing angle, on the other hand, can only be influenced by changing the size or distance of the object to be recorded. In trick technology and architectural photography, for example, the viewing angle is modified by instead of the object to be depicted - e.g. B. of a building or architectural details - a reduced (or enlarged) model of the same photographed or filmed. The size of “real” objects, on the other hand, cannot normally be influenced - the only thing left to the photographer here is usually to choose a different distance to the object, also jokingly referred to as “ foot zoom ”.

### Sample table

 Focal length / mm Angle of view APS-C / degree Full frame angle of view / degree Angle of view APS-C / rad Angle of view full format / rad 0 not defined not defined not defined not defined 1 171.5 174.7 2.9926 3.0493 5 139.1 154.0 2.4273 2.6877 10 106.5 130.4 1.8594 2.2762 20th 67.6 94.5 1.1806 1.6500 30th 48.1 71.6 0.8402 1.2502 40 37.0 56.8 0.6465 0.9922 50 30.0 46.8 0.5237 0.8173 60 25.2 39.7 0.4395 0.6926 70 21.7 34.4 0.3783 0.5999 80 19.0 30.3 0.3319 0.5286 100 15.3 24.4 0.2664 0.4264 200 7.7 12.4 0.1338 0.2157 1000 1.5 2.5 0.0268 0.0433 10,000 0.15 0.25 0.0027 0.0043 100,000 0.02 0.02 0.0003 0.0004

## Macro photography

The definition of the angle of view relates to the focal plane . However, this does not necessarily coincide with the image plane . If an object is to be shown sharply on a film or a focusing screen at a short distance, it must either be dimmed or the image plane shifted backwards. If the object distance is not greater than twice the focal length, one speaks of macro photography .

In order to obtain the effective angle of view - in relation to the image plane  - in the close range, the image distance must be used in the definition instead of the focal length , i.e. ${\ displaystyle \ textstyle b}$

${\ displaystyle \ alpha _ {\ text {eff}} = 2 \ cdot \ arctan \ left ({\ frac {d} {2 \ cdot b}} \ right).}$

Expressed by the focal length and the object distance, the lens equation gives : ${\ displaystyle \ textstyle f}$${\ displaystyle \ textstyle g}$

${\ displaystyle \ alpha _ {\ text {eff}} = 2 \ cdot \ arctan \ left ({\ frac {d} {2 \ cdot f}} - {\ frac {d} {2 \ cdot g}} \ right) = 2 \ cdot \ arctan \ left ({\ frac {d} {2}} \ cdot \ left ({\ frac {1} {f}} - {\ frac {1} {g}} \ right) \ right).}$

Note: The
same applies to other basic definitions in photography. In the macro area, one speaks analogously of effective aperture or effective light intensity , effective magnification or also effective exposure time , which result from replacing the focal length with the image distance.