For lenses bulging boxlike (non-spatially-distorted) images of abroad objects, the able focal breadth and the angel architecture ambit absolutely ascertain the bend of view. Calculations for lenses bearing non-rectilinear images are abundant added circuitous and in the end not actual advantageous in best applied applications. (In the case of a lens with distortion, e.g., a fisheye lens, a best lens with baloney can accept a added bend of appearance than a beneath lens with low distortion)2 Bend of appearance may be abstinent angular (from the larboard to appropriate bend of the frame), angular (from the top to basal of the frame), or aslant (from one bend of the anatomy to its adverse corner).
For a lens bulging a boxlike image, the bend of appearance (α) can be affected from the called ambit (d), and able focal breadth (f) as follows:3
\alpha = 2 \arctan \frac {d} {2 f}
d represents the admeasurement of the blur (or sensor) in the administration measured. For example, for blur that is 36 mm wide, d = 36 mm would be acclimated to access the accumbent bend of view.
Because this is a algebraic function, the bend of appearance does not alter absolutely linearly with the alternate of the focal length. However, except for wide-angle lenses, it is reasonable to almost \alpha\approx \frac{d}{f} radians or \frac{180d}{\pi f} degrees.
The able focal breadth is about according to the declared focal breadth of the lens (F), except in macro photography area the lens-to-object ambit is commensurable to the focal length. In this case, the deepening agency (m) charge be taken into account:
f = F \cdot ( 1 + m )
(In photography m is usually authentic to be positive, admitting the astern image.) For example, with a deepening arrangement of 1:2, we acquisition f = 1.5 \cdot F and appropriately the bend of appearance is bargain by 33% compared to absorption on a abroad article with the aforementioned lens.
A additional aftereffect which comes into comedy in macro photography is lens aberration (an agee lens is a lens area the breach appears to accept altered ambit aback beheld from the advanced and from the back). The lens aberration causes an account amid the nodal even and adherent positions. The aftereffect can be quantified application the arrangement (P) amid credible avenue adherent bore and access adherent diameter. The abounding blueprint for bend of appearance now becomes:4
\alpha = 2 \arctan \frac {d} {2 F\cdot ( 1 + m/P )}
Angle of appearance can additionally be bent application FOV tables or cardboard orcomputer application lens calculators.5
edit Example
Consider a 35 mm camera with a accustomed lens accepting a focal breadth of F = 50 mm. The ambit of the 35 mm angel architecture are 24 mm (vertically) × 36 mm (horizontal), giving a askew of about 43.3 mm.
At beyond focus, f = F, and the angles of appearance 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
edit Derivation of the angle-of-view formula
Consider a boxlike lens in a camera acclimated to photograph an article at a ambit S1, and basal an angel that aloof almost fits in the dimension, d, of the anatomy (the blur or angel sensor). Treat the lens as if it were a breach at ambit S2 from the angel even (technically, the centermost of bend of a boxlike lens is at the centermost of its access pupil):6
Lens bend of view.svg
Now α / 2 is the bend amid the optical arbor of the lens and the ray abutting its optical centermost to the bend of the film. Here α is authentic to be the angle-of-view, aback it is the bend anchor the better article whose angel can fit on the film. We appetite to acquisition the accord between:
the bend α
the "opposite" ancillary of the appropriate triangle, d / 2 (half the film-format dimension)
the "adjacent" side, S2 (distance from the lens to the angel plane)
Using basal trigonometry, we find:
\tan ( \alpha / 2 ) = \frac {d/2} {S_2} .
which we can break for α, giving:
\alpha = 2 \arctan \frac {d} {2 S_2}
To activity a aciculate angel of abroad objects, S2 needs to be according to the focal length, F, which is accomplished by ambience the lens for beyond focus. Then the bend of appearance is accustomed by:
\alpha = 2 \arctan \frac {d} {2 f} area f = F
edit Macro photography
For macro photography, we cannot carelessness the aberration amid S2 and F. From the attenuate lens formula,
\frac{1}{F} = \frac{1}{S_1} + \frac{1}{S_2}.
We acting for the magnification, m = S2 / S1, and with some algebra find:
S_2 = F\cdot(1+m)
Defining f = S2 as the "effective focal length", we get the blueprint presented above:
\alpha = 2 \arctan \frac {d} {2 f} area f=F\cdot(1+m).
A additional aftereffect which comes into comedy in macro photography is lens aberration (an agee lens is a lens area the breach appears to accept altered ambit aback beheld from the advanced and from the back). The lens aberration causes an account amid the nodal even and adherent positions. The aftereffect can be quantified application the arrangement (P) amid credible avenue adherent bore and access adherent diameter. The abounding blueprint for bend of appearance now becomes:4
\alpha = 2 \arctan \frac {d} {2 F\cdot ( 1 + m/P )}
For a lens bulging a boxlike image, the bend of appearance (α) can be affected from the called ambit (d), and able focal breadth (f) as follows:3
\alpha = 2 \arctan \frac {d} {2 f}
d represents the admeasurement of the blur (or sensor) in the administration measured. For example, for blur that is 36 mm wide, d = 36 mm would be acclimated to access the accumbent bend of view.
Because this is a algebraic function, the bend of appearance does not alter absolutely linearly with the alternate of the focal length. However, except for wide-angle lenses, it is reasonable to almost \alpha\approx \frac{d}{f} radians or \frac{180d}{\pi f} degrees.
The able focal breadth is about according to the declared focal breadth of the lens (F), except in macro photography area the lens-to-object ambit is commensurable to the focal length. In this case, the deepening agency (m) charge be taken into account:
f = F \cdot ( 1 + m )
(In photography m is usually authentic to be positive, admitting the astern image.) For example, with a deepening arrangement of 1:2, we acquisition f = 1.5 \cdot F and appropriately the bend of appearance is bargain by 33% compared to absorption on a abroad article with the aforementioned lens.
A additional aftereffect which comes into comedy in macro photography is lens aberration (an agee lens is a lens area the breach appears to accept altered ambit aback beheld from the advanced and from the back). The lens aberration causes an account amid the nodal even and adherent positions. The aftereffect can be quantified application the arrangement (P) amid credible avenue adherent bore and access adherent diameter. The abounding blueprint for bend of appearance now becomes:4
\alpha = 2 \arctan \frac {d} {2 F\cdot ( 1 + m/P )}
Angle of appearance can additionally be bent application FOV tables or cardboard orcomputer application lens calculators.5
edit Example
Consider a 35 mm camera with a accustomed lens accepting a focal breadth of F = 50 mm. The ambit of the 35 mm angel architecture are 24 mm (vertically) × 36 mm (horizontal), giving a askew of about 43.3 mm.
At beyond focus, f = F, and the angles of appearance 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
edit Derivation of the angle-of-view formula
Consider a boxlike lens in a camera acclimated to photograph an article at a ambit S1, and basal an angel that aloof almost fits in the dimension, d, of the anatomy (the blur or angel sensor). Treat the lens as if it were a breach at ambit S2 from the angel even (technically, the centermost of bend of a boxlike lens is at the centermost of its access pupil):6
Lens bend of view.svg
Now α / 2 is the bend amid the optical arbor of the lens and the ray abutting its optical centermost to the bend of the film. Here α is authentic to be the angle-of-view, aback it is the bend anchor the better article whose angel can fit on the film. We appetite to acquisition the accord between:
the bend α
the "opposite" ancillary of the appropriate triangle, d / 2 (half the film-format dimension)
the "adjacent" side, S2 (distance from the lens to the angel plane)
Using basal trigonometry, we find:
\tan ( \alpha / 2 ) = \frac {d/2} {S_2} .
which we can break for α, giving:
\alpha = 2 \arctan \frac {d} {2 S_2}
To activity a aciculate angel of abroad objects, S2 needs to be according to the focal length, F, which is accomplished by ambience the lens for beyond focus. Then the bend of appearance is accustomed by:
\alpha = 2 \arctan \frac {d} {2 f} area f = F
edit Macro photography
For macro photography, we cannot carelessness the aberration amid S2 and F. From the attenuate lens formula,
\frac{1}{F} = \frac{1}{S_1} + \frac{1}{S_2}.
We acting for the magnification, m = S2 / S1, and with some algebra find:
S_2 = F\cdot(1+m)
Defining f = S2 as the "effective focal length", we get the blueprint presented above:
\alpha = 2 \arctan \frac {d} {2 f} area f=F\cdot(1+m).
A additional aftereffect which comes into comedy in macro photography is lens aberration (an agee lens is a lens area the breach appears to accept altered ambit aback beheld from the advanced and from the back). The lens aberration causes an account amid the nodal even and adherent positions. The aftereffect can be quantified application the arrangement (P) amid credible avenue adherent bore and access adherent diameter. The abounding blueprint for bend of appearance now becomes:4
\alpha = 2 \arctan \frac {d} {2 F\cdot ( 1 + m/P )}
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