Angle of view
In photography, angle of view describes the angular extent of a given scene that is imaged by a camera. It is used interchangeably with the more general term field of view.
It is important to distinguish the angle of view from the angle of coverage, which describes the angle of projection by the lens onto the focal plane. For most cameras, it may be assumed that the image circle produced by the lens is large enough to cover the film or sensor completely.One exception is technical photography involving view camera movements, in which the optical axis of the lens may not be aligned with the center of the frame. The photographer must ensure that the angle of coverage of the lens is large enough to cover the frame in this case. If the angle of view exceeds the angle of coverage, however, then vignetting will be present in the resulting photograph.
Calculating a camera's angle of view
For lenses projecting rectilinear (non-spatially-distorted) images of distant objects, the effective focal length and the image format dimensions completely define the angle of view. Calculations for lenses producing non-rectilinear images are much more complex and in the end not very useful in most practical applications. Angle of view may be measured horizontally (from the left to right edge of the frame), vertically (from the top to bottom of the frame), or diagonally (from one corner of the frame to its opposite corner). For a lens projecting a rectilinear image, the angle of view (α) can be calculated from the chosen dimension (d), and effective focal length (f) as follows: :\alpha = 2 \arctan \frac {d} {2 f} d represents the size of the film (or sensor) in the direction measured. For example, for film that is 36 mm wide, d = 36 mm would be used to obtain the horizontal angle of view. Because this is a trigonometric function, the angle of view does not vary quite linearly with the reciprocal of the focal length. However, except for wide-angle lenses, it is reasonable to approximate \alpha\approx \frac{d}{f} radians or \frac{180d}{\pi f} degrees. The effective focal length is nearly equal to the stated focal length of the lens (F), except in macro photography where the lens-to-object distance is comparable to the focal length. In this case, the magnification factor (m) must be taken into account: :f = F \cdot ( 1 + m ) (In photography m is usually defined to be positive, despite the inverted image.) For example, with a magnification ratio of 1:2, we find f = 1.5 \cdot F and thus the angle of view is reduced by 33% compared to focusing on a distant object with the same lens. 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. CCTV Field of View Camera Lens Calculations by JVSG, December, 2007Example
Consider a 35 mm camera with a normal lens having a focal length of . 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, , 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=FMacro 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}. We substitute for the magnification, m = S_2/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
In the optical instrumentation industry the term field of view (FOV) is most often used, though the measurements are still expressed as angles.Holst, G.C. (1998). Testing and Evaluation of Infrared Imaging Systems (2nd ed.). Florida:JCD Publishing, Washington:SPIE. 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.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 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.Electro Optical Industries, Inc.(2005). EO TestLab Methadology. In Education/Ref. http://www.electro-optical.com/html/toplevel/educationref.asp. 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
- Ultra wide angle lenses (less than 24mm of focal length in 35mm film format), also known as fisheye lenses if not rectilinear, cover up to 180° (or even wider in special cases)
- *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.
- Wide-angle lenses (24-35mm) cover between 84° and 64°
- Normal, or Standard lenses (36-60mm) cover between 62° and 40°
- Telephoto lenses generally cover between 30° and 10°
- Super Telephoto lenses generally cover between 8° through less than 1°