視角

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

在攝影學中，視角(angle of view)是在一般環境中，相機可以接收影像的角度範圍，也可以常被稱為視野。

視角(angle of view)與成像範圍(angle of coverage)是不同的，他是描述鏡頭可以擷取的影像角度，一般來說鏡頭的成像圈都夠大到涵蓋底片或者感光元件（或許會有一點點的邊緣暗角）。假如鏡頭的成像範圍無法涵蓋整個感光元件，則成像圈會被看見，一般會伴隨嚴重的邊緣暗角，在這個狀態下，視角會被成像範圍所限制。

In 1916, Northey showed how to calculate the angle of view using ordinary carpenter's tools. The angle that he labels as the angle of view is the half-angle or "the angle that a straight line would take from the extreme outside of the field of view to the center of the lens;" he notes that manufacturers of lenses use twice this angle.

計算一台相機的視角

針對直線投射鏡頭（無空間扭曲）產生的遙遠物體影像，有效焦距與影像格式尺寸足以定義視角。 計算非線性影像相對複雜許多，而且在大部分的實際應用上並不是非常有用。（在透鏡扭曲的的情況下，譬如魚眼鏡頭，有扭曲的長段鏡頭可以比較低扭曲的短鏡頭有較寬的視角） 視角也許可以用水平（從影像的左端至右端），垂直（從影像頂端至底端）或者斜角(從影像一角至對角)等方式計算出來。

對於直線投射影像，視角 (α)可以由被選擇的大小(d),以及有效焦段(f)計算出來如下：

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

$d$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。

有效焦距趨近等於標示的鏡頭焦距(F)，除了微距攝影鏡頭至拍攝物距離與焦距接近。在此案例中，放大倍率(m) 必須加以考慮：

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

(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%。

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:

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

Angle of view can also be determined using FOV tables or paper or software lens calculators.

Example

Consider a 35 mm camera with a lens having a focal length of F = 50 mm. The dimensions of the 35 mm image format are 24 mm (vertically) × 36 mm (horizontal), giving a diagonal of about 43.3 mm.

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):

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:

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

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. 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.

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.

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) generally have an angle of view of 35° or less. Since photographers usually only encounter the telephoto lens sub-type, 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°
• "Super telephoto" (over 300mm in 35mm film format) generally cover between 8° through less than 1°

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.

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). Digital compact cameras sometimes state the focal lengths of their lenses in 35mm equivalents, which can be used in this table.

Focal Length (mm)	0	2	12	14	16	20	24	35	50	70	85	100	200	300	400	500	600	700	800	1200
Diagonal (°)	180.0	169.4	122.0	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	3.10	2.07
Vertical (°)	180.0	161.1	90.0	81.2	73.9	61.9	53.1	37.8	27.0	19.5	16.1	13.0	6.87	4.58	3.44	2.75	2.29	1.96	1.72	1.15
Horizontal (°)	180.0	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

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.

See also

• Simple Explanation of Angle of View and Focal Length
• 35 mm equivalent focal length
• Camera angle
• Camera coverage
• Camera operator
• Cinematic techniques
• Field of view
• Filmmaking
• Multiple-camera setup
• Single-camera setup
• Video production

External links

• Angle of View on digital SLR cameras with reduced sensor size
• Focal Length and Angle of View