# 視角

A camera's angle of view can be measured horizontally, vertically, or diagonally.

## 計算一台相機的視角

$\alpha = 2 \arctan \frac {d} {2 f}$

$d$表示底片(或感光元件)的大小再一個方向的計算。譬如，對於36mm寬的底片，$d = 36$ mm 可以被拿來取得水平視角。 由於這是三角函數方程式，視角不會與焦距呈現性關係。然而，除了寬角度鏡頭，他是合理的近似$\alpha\approx \frac{d}{f}$ 弧度或 $\frac{180d}{\pi f}$ 角度。

$f = F \cdot ( 1 + m )$

(在攝影學中 $m$ 通常被定義為正值，儘管是被顛倒的影像。) 舉例來說， 在放大倍率1:2的狀況下， 我們發現 $f = 1.5 \cdot F$ 而與一個對遠處的物體具有相同焦距相比視角減少了33%。

$\alpha = 2 \arctan \frac {d} {2 F\cdot ( 1 + m/P )}$

### 示例

At infinity focus, f = F, and the angles of view are:

• horizontally, $\alpha_h = 2\arctan\frac{h}{2f} = 2\arctan\frac{36}{2 \times 50}\approx 39.6^\circ$
• vertically, $\alpha_v = 2\arctan\frac{v}{2f} = 2\arctan\frac{24}{2 \times 50}\approx 27.0^\circ$
• diagonally, $\alpha_d = 2\arctan\frac{d}{2f} = 2\arctan\frac{43.3}{2 \times 50}\approx 46.8^\circ$

### Derivation of the angle-of-view formula

Consider a rectilinear lens in a camera used to photograph an object at a distance $S_1$, and forming an image that just barely fits in the dimension, $d$, of the frame (the film or image sensor). Treat the lens as if it were a pinhole at distance $S_2$ from the image plane (technically, the center of perspective of a rectilinear lens is at the center of its entrance pupil):[5]

Now $\alpha/2$ is the angle between the optical axis of the lens and the ray joining its optical center to the edge of the film. Here $\alpha$ is defined to be the angle-of-view, since it is the angle enclosing the largest object whose image can fit on the film. We want to find the relationship between:

the angle $\alpha$
the "opposite" side of the right triangle, $d/2$ (half the film-format dimension)
the "adjacent" side, $S_2$ (distance from the lens to the image plane)

Using basic trigonometry, we find:

$\tan ( \alpha / 2 ) = \frac {d/2} {S_2} .$

which we can solve for α, giving:

$\alpha = 2 \arctan \frac {d} {2 S_2}$

To project a sharp image of distant objects, $S_2$ needs to be equal to the focal length, $F$, which is attained by setting the lens for infinity focus. Then the angle of view is given by:

$\alpha = 2 \arctan \frac {d} {2 f}$ where $f=F$

#### Macro photography

For macro photography, we cannot neglect the difference between $S_2$ and $F$. From the thin lens formula,

$\frac{1}{F} = \frac{1}{S_1} + \frac{1}{S_2}$.

From the definition of magnification, $m = S_2/S_1$, we can substitute $S_1$ and with some algebra find:

$S_2 = F\cdot(1+m)$

Defining $f=S_2$ as the "effective focal length", we get the formula presented above:

$\alpha = 2 \arctan \frac {d} {2 f}$ where $f=F\cdot(1+m)$.

A second effect which comes into play in macro photography is lens asymmetry (an asymmetric lens is a lens where the aperture appears to have different dimensions when viewed from the front and from the back). The lens asymmetry causes an offset between the nodal plane and pupil positions. The effect can be quantified using the ratio (P) between apparent exit pupil diameter and entrance pupil diameter. The full formula for angle of view now becomes:[3]

$\alpha = 2 \arctan \frac {d} {2 F\cdot ( 1 + m/P )}$

## Measuring a camera's field of view

File:FOV test Optics apparatus.PNG
Schematic of collimator-based optical apparatus used in measuring the FOV of a camera.

In the optical instrumentation industry the term field of view (FOV) is most often used, though the measurements are still expressed as angles.[6] Optical tests are commonly used for measuring the FOV of UV, visible, and infrared (wavelengths about 0.1–20 µm in the electromagnetic spectrum) sensors and cameras.

The purpose of this test is to measure the horizontal and vertical FOV of a lens and sensor used in an imaging system, when the lens focal length or sensor size is not known (that is, when the calculation above is not immediately applicable). Although this is one typical method that the optics industry uses to measure the FOV, there exist many other possible methods.

UV/visible light from an integrating sphere (and/or other source such as a black body) is focused onto a square test target at the focal plane of a collimator (the mirrors in the diagram), such that a virtual image of the test target will be seen infinitely far away by the camera under test. The camera under test senses a real image of the virtual image of the target, and the sensed image is displayed on a monitor.[7]

File:FOV Target on Monitor.png
Monitor display of sensed image from the camera under test

The sensed image, which includes the target, is displayed on a monitor, where it can be measured. Dimensions of the full image display and of the portion of the image that is the target are determined by inspection (measurements are typically in pixels, but can just as well be inches or cm).

$D$ = dimension of full image
$d$ = dimension of image of target

The collimator's distant virtual image of the target subtends a certain angle, referred to as the angular extent of the target, that depends on the collimator focal length and the target size. Assuming the sensed image includes the whole target, the angle seen by the camera, its FOV, is this angular extent of the target times the ratio of full image size to target image size.[8]

The target's angular extent is:

$\alpha = 2 \arctan \frac {L} {2 f_c}$
where $L$ is the dimension of the target and $f_c$ is the focal length of collimator.

The total field of view is then approximately:

$\mathrm{FOV} = \alpha \frac{D}{d}$

or more precisely, if the imaging system is rectilinear:

$\mathrm{FOV} = 2 \arctan \frac {LD} {2 f_c d}$

This calculation could be a horizontal or a vertical FOV, depending on how the target and image are measured.

## Lens types and effects

### Focal length

How focal length affects perspective: Varying focal lengths at identical field size achieved by different camera-subject distances. Notice that the shorter the focal length and the larger the angle of view, perspective distortion and size differences increase.

Lenses are often referred to by terms that express their angle of view:

• fisheye lenses, typical focal lengths are between 8 mm and 10 mm for circular images, and 15–16 mm for full-frame images. Up to 180° and beyond.
• A circular fisheye lens (as opposed to a full-frame fisheye) is an example of a lens where the angle of coverage is less than the angle of view. The image projected onto the film is circular because the diameter of the image projected is narrower than that needed to cover the widest portion of the film.
• Ultra wide angle lens is a rectilinear which is less than 24mm of focal length in 35mm film format, here 14mm is 114° and 24mm is 84° .
• Wide-angle lenses (24–35mm in 35mm film format) cover between 84° and 64°
• Normal, or Standard lenses (36–60mm in 35mm film format) cover between 62° and 40°
• Long focus lenses (any lens with a focal length greater than the diagonal of the film or sensor used)[9] generally have an angle of view of 35° or less.[10] Since photographers usually only encounter the telephoto lens sub-type,[11] they are referred to in common photographic parlance as:
• "Medium telephoto", a focal length of 85mm to 135mm in 35mm film format covering between 30° and 10°[12]
• "Super telephoto" (over 300mm in 35mm film format) generally cover between 8° through less than 1°[12]

Zoom lenses are a special case wherein the focal length, and hence angle of view, of the lens can be altered mechanically without removing the lens from the camera.

### Characteristics

For a given camera–subject distance, longer lenses magnify the subject more. For a given subject magnification (and thus different camera–subject distances), longer lenses appear to compress distance; wider lenses appear to expand the distance between objects.

Another result of using a wide angle lens is a greater apparent perspective distortion when the camera is not aligned perpendicularly to the subject: parallel lines converge at the same rate as with a normal lens, but converge more due to the wider total field. For example, buildings appear to be falling backwards much more severely when the camera is pointed upward from ground level than they would if photographed with a normal lens at the same distance from the subject, because more of the subject building is visible in the wide-angle shot.

Because different lenses generally require a different camera–subject distance to preserve the size of a subject, changing the angle of view can indirectly distort perspective, changing the apparent relative size of the subject and foreground.

If the subject image size remains the same, then at any given aperture all lenses, wide angle and long lenses, will give the same depth of field.[13]

### Examples

An example of how lens choice affects angle of view. The photos below were taken by a 35 mm still camera at a constant distance from the subject:

 28 mm lens, 65.5° × 46.4° 50 mm lens, 39.6° × 27.0° 70 mm lens, 28.9° × 19.5° 210 mm lens, 9.8° × 6.5°

## Common lens angles of view

This table shows the diagonal, horizontal, and vertical angles of view, in degrees, for lenses producing rectilinear images, when used with 36 mm × 24 mm format (that is, 135 film or full-frame 35mm digital using width 36 mm, height 24 mm, and diagonal 43.3 mm for d in the formula above).[14] Digital compact cameras sometimes state the focal lengths of their lenses in 35mm equivalents, which can be used in this table.

 Focal Length (mm) Diagonal (°) Vertical (°) Horizontal (°) 0 2 12 14 16 20 24 35 50 70 85 100 200 300 400 500 600 700 800 1200 180 169.4 122 114.2 107.1 94.5 84.1 63.4 46.8 34.4 28.6 23.3 12.4 8.25 6.19 4.96 4.13 5.72 3.1 2.07 180 161.1 90 81.2 73.9 61.9 53.1 37.8 27 19.5 16.1 13 6.87 4.58 3.44 2.75 2.29 1.96 1.72 1.15 180 166.9 111.1 102.7 95.1 82.4 73.7 54.4 39.6 28.8 23.9 19.5 10.3 6.87 5.15 4.12 2.86 3.54 2.58 1.72
Five images using 24, 28, 35, 50 and 72mm equivalent zoom lengths, portrait format, to illustrate angles of view [15]
Five images using 24, 28, 35, 50 and 72mm equivalent step zoom function, to illustrate angles of view

## Three-dimensional digital art

Displaying 3d graphics as 3d projection of the models onto a 2d surface uses a series of mathematical calculations to render the scene. The angle of view of the scene is thus readily set and changed; some renderers even measure the angle of view as the focal length of an imaginary lens. The angle of view can also be projected onto the surface at an angle greater than 90°, effectively creating an ultra-wide lens effect.

## Cinematography and video gaming

Modifying the angle of view over time, or zooming, is a frequently used cinematic technique.

For a visual effect, some first person video games (especially racing games), widen the angle of view beyond 90° to exaggerate the distance the player is travelling, thus exaggerating the player's perceived speed and giving a tunnel effect (like pincushion distortion). Narrowing the view angle gives a zoom in effect. Also see Field of view in video games.

## References and notes

1. ^ http://www.the-digital-picture.com/reviews/canon-ef-15mm-f-2.8-fisheye-lens-review.aspx
2. ^ Ernest McCollough. Photographic Topography. Industry: A Monthly Magazine Devoted to Science, Engineering and Mechanic Arts (Industrial Publishing Company, San Francisco). 1893: 399–406.
3. ^ 3.0 3.1 Paul van Walree. Center of perspective. 2009 [24 Januari 2010].
4. ^ CCTV Field of View Camera Lens Calculations by JVSG, December, 2007
5. ^ Kerr, Douglas A. The Proper Pivot Point for Panoramic Photography (PDF). The Pumpkin. 2005 [2007-01-14].
6. ^ Holst, G.C. (1998). Testing and Evaluation of Infrared Imaging Systems (2nd ed.). Florida:JCD Publishing, Washington:SPIE.
7. ^ Mazzetta, J.A.; Scopatz, S.D. (2007). Automated Testing of Ultraviolet, Visible, and Infrared Sensors Using Shared Optics. Infrared Imaging Systems: Design Analysis, Modeling, and Testing XVIII,Vol. 6543, pp. 654313-1 654313-14
8. ^ Electro Optical Industries, Inc.(2005). EO TestLab Methadology. In Education/Ref. http://www.electro-optical.com/html/toplevel/educationref.asp.
9. ^ Sidney F. Ray, Applied photographic optics: lenses and optical systems for photography, page 294
10. ^ Lynne Warren, Encyclopedia of 20th century photography, page 211
11. ^ Michael Langford, Basic photography, page 83
12. ^ 12.0 12.1 photographywebsite.co.uk - Lens Types Explained
13. ^ Reichmann, Michael. Do Wide Angle Lenses Really Have Greater Depth of Field Than Telephotos?.
14. ^ However, most interchangeable-lens digital cameras do not use 24x36 mm image sensors and therefore produce narrower angles of view than set out in the table. See crop factor and the subtopic digital camera issues in the article on wide-angle lenses for further discussion.
15. ^ The image examples uses a 5.1-15.3mm lens which is called a 24mm 3x zoom by the producer (Ricoh Caplio GX100)

## 外部連結

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