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Math of Field of View (FOV) for a Camera and Lens

There are approximations in calculations. The math is precise, but the data is less so. The required Focal length and Sensor Size are rounded specifications, losing a bit of precision. This little difference at the small sensor gets magnified in the field and scene. However, the results certainly are close enough to be very useful in any practical case. My experience is that the field is fairly accurate (at distances of at least a meter or so), assuming you actually know your parameters. Some problems are:

Thin Lens Equation

The Marked focal length of any lens is a rounded nominal number, like 50 or 60 mm. The actual can be a few percent different. Furthermore, the Marked focal length is only applicable to focus at infinity. Focal length normally increases when lens is extended forward to focus closer. Lens specs normally indicate this internal extension at closest focus distance. But internal focusing lenses can do internal tricks with focal length (some zooms and macro lenses can be shorter when up close, instead of longer). But generally speaking, focal length becomes a little longer at very close distances (2x longer at 1:1), so field of view could be a little smaller, but should be insignificant as long as magnification is greater than 0.1x, which in regular lenses is normally near their Minimum focus distance. But this calculation does not include macro distances. We are only seeking a ballpark FOV number anyway, we adjust small differences with our subject framing or cropping, but vague guesses about your distance and sensor size or focal length don't help FOV accuracy.

And a fisheye lens is a different animal, wider view than this formula predicts. A regular lens is rectilinear, meaning it shows straight lines as straight lines, not curved. A fisheye is rather unconcerned about this distortion, and can show a wider view, poorly purists might say, but very wide, and very possibly interesting.

Focal Length varies with zoom, and the Exif data with the image should report focal length (in coarse zoom steps, as best it can). Actual focal length could be determined by the Magnification (Wikipedia) and distance. The distance from the front nodal point to the object in the subject plane (s1), and the distance from the rear nodal point to the image plane (s2) (when focused) are related by this Thin Lens equation (Wikipedia), which the diagram and formulas below simplify. If OK with a little geometry and algebra, you can see the derivation of this classic Thin Lens Equation at the Khan Academy.

In this equation, we can see that if the subject at s1 is at infinity, then 1/s1 is zero, so then s2 = f. This is the marked focal length that applies when focused at infinity. What camera lenses call the focused distance is s = s1 + s2, from subject to the sensor plane.

Also if at 1:1 magnification (equal image size on both sides of lens), then s1 = s2, saying that the working macro distance in front of the lens node (extended at 1:1) is equal to the distance to the sensor image plane (both sides at 2x focal length). This makes f/stop number be 2x, which is 2 EV light loss. Those are basics. Internal focusing macro lenses can slightly reduce this light loss, but it is still near 2 EV.


The Math

fov

This Thin Lens Model simplifies, as if it were a simple one glass element (like a handheld magnifying lens) with only the one central node point. Multi-element “Thick” camera lenses have two nodes for each side of lens (see the fstop page). But camera lenses normally have several (to many) glass lens elements (a thick lens), designed to correct optical aberrations and distortions, and also to zoom. The Thin Lens Model does work well for practical computing purposes (a bit more description below).

The focal length is measured from the sensor plane to the lens node H' (often inside the lens, but not always). Technically, the focal length Marked on the lens applies when focused at infinity, but it will be a bit longer when focused closer. Lens equations use distance d in front of the field node, however note that specifications of lens focus-distance (like minimum closest focus) specify s to the sensor, the sum of d and f.

The standard camera magnification geometry uses the standard ratios of the similar triangles shown above. The field dimension angle in front of this lens node is the same angle (opposite angles) as the sensor dimension angle behind the lens. The ratio of distances on each side of the lens are the same as the ratio of the size dimensions on each side of the lens. These ratios (as shown first below) are simply the trigonometry tangents of the same angle on each side of the lens (tangent is opposite side over adjacent side). In this equation, rearrangement will compute any one term from the other three. But one mm of error in focal length or sensor size is magnified in the field, so input accuracy is critical. The math is simple, but the difficult part of computing this will be to first accurately determine the correct sensor dimensions, and/or the zoom lens focal length (small errors get magnified at the larger distant field).

Trigonometry does also work for this, but we don't need trig except to compute the actual angles of Field of View. Because Field of View dimensions are just similar triangles (shown gray above). The three field or sensor dimensions are Height, Width, or Diagonal, each are computed individually. Using the (opposite dimension / 2) to create right angles for trig, these ratios are the trig tangent of the half angles, which is the same opposite angle on both sides of lens (the two formula ratios are necessarily equal). The /2 cancels out on both sides, and any unit conversions to feet or meters on the right side cancel out. It becomes a very simple equation. Do be consistent with units, but then there is no conversion of units needed (same ratio on both sides).

Basics of lens optics in all of these equations (similar triangles on both sides of the lens):

Sensor dimension (mm)
Focal length (mm)
  = 
Field dimension *
Distance to Field *

* feet or meters (but both same units)

Meaning, if you use Feet for Distance, then the Field Width will also be feet. Or meters if you use meters.

Rearrange ratios to compute desired unknown value. The actual Field of View Calculator will be much more versatile, and can help find sensor size from crop factor, but math examples are shown here for Width. The computed Dimension can be Width, Height, or Diagonal.

Sensor Width mm × Distance
Focal Length mm
  = Field Width  

Field Width × Focal Length mm
Sensor Width mm
  = Distance   

This looks simple because it is (but correct sensor size and focal length are required). Twice the focal length is half the field, or twice the distance is twice the field dimension. And assuming use of a proper lens designed for the sensor, twice the sensor dimension is twice the field, which is why a smaller "cropped" sensor sees a smaller field unless compensated with a shorter lens.

These ratios above are just the trig tangent (opposite over adjacent), necessarily equal for the equal angles, but using the ratios is simpler math for the field dimensions.

But computing the angle of view requires trigonometry, which is again, Width, Height, or Diagonal.

Field angle of view = 2 x arctan ((sensor dimension (mm) / 2) / focal length (mm))

2 × arctan(
Sensor Width mm
2 × Focal Length mm
)  =  Degrees  

Trig functions in software programming languages (including Excel, Javascript, C, Python, etc.) use radians instead of degrees. Calculators normally offer the choice. To convert radians to degrees, multiply radians by 180/Pi. which is 57.2958 degrees per radian. Or for degrees to radians, multiply degrees by Pi/180.

Field size and focal length are linear in field of view (half of sensor dimension or twice the focal length both produce half the field dimension). But the angle is a tangent function, which is Not straight-line linear (the angle is approximately linear if angle is not more than about 10 degrees (called the Small-Angle Approximation).


Substituting size of an Object in the field, instead of overall Field size. Using these two steps may be convenient:

Object height on sensor (mm)
  =  
Sensor height (mm) × Object height (pixels)

Sensor height (pixels)

Object height on sensor (mm)
Focal length (mm)
  =  
Real Object size *
Distance to Object *


Rearrange this ratio to compute desired unknown value. Examples:

Distance to Object *
  =  
Real Object height * × Focal Length (mm)

Object height on sensor (mm)

Real Object height *
  =  
Distance to Object * × Object height on sensor (mm)

Focal Length (mm)

* feet or meters (but both same units)


There are Other possible arrangements. But this next one must use Field dimension and Height converted to mm (304.8 mm per foot), because each ratio must be a dimensionless number.

Magnification
Distance to Object (mm)

Focal Length (mm)
Real Object height (mm)

Object height on sensor (mm)

This is magnification on the sensor or film. The image must still be enlarged for viewing.
1:1 macro reproduction at 1× is when these two top and bottom values are equal.
Distant object size is reduced greatly, like perhaps maybe magnification of 0.001x or less.


Back to the general ideas, of all above:

This Distance to Field doesn't necessarily mean to subject or to focus point. Here it means the distance to the point where you want field size calculated (perhaps the background, computing what will show in the picture). As noted, use either meters or feet (the units all cancel out). And/or substitute width for height if appropriate. Just be consistent, and solve for the unknown. FWIW, cameras report focus distance as s, measured to the sensor surface. Lens calculations however compute with distance d, in front of the lens node (which except for telephoto lenses, is normally inside the lens body).

The Thin Lens Equations use the distance d in front of the lens node, but the lens specifications are "focused distance" (marked s in the diagram) which is to the sensor plane.

Working Distance = d = S1 (distance in front of lens)
Focal Length = f = S2 (distance behind lens)
Focus Distance = d + f

The Thin Lens Model simplifies, and is practical and adequate for most computing, even if this model has one glass element and one central node point. Camera lenses have multiple glass elements, and are "thick lenses", much more complex, and have two node points, often called H for the field view side node, and H' for the sensor view side node. These two nodes might be designed a few inches apart, but they necessarily still see the same angle. The actual node position normally only matters to us for close macro distances, or perhaps in efforts to correct panoramic parallax.

This front node is often a point inside the lens, perhaps crudely assumed about the middle of the lens (but it does move with zoom), but it can often be ignored. Lens specifications normally instead measure field focus distance from the sensor, so you could subtract a few inches from your subject distance (to be at that node) to compute Field of View, but normally this is not significant except at close minimum focus distance and macro distances. But it is very significant for macro distances. Example, the Nikon 105mm f/2.8 VR macro lens has specification: Minimum focus distance: 31.4 cm (1.03 feet), which is measured to the sensor plane at rear of camera (there's a small symbol on top of the camera there, much like Greek letter Theta ϴ, to mark the sensor plane location if measuring.) But at 1:1, a reliable chart of macro lens working distances says that this Working Distance (in front of lens in practice, but technically designed to the H node) at 1:1 is significantly less, 14.8 cm (5.8 inches), which is a difference of the length of the lens and thickness of the camera body (sensor to mounting flange).

Distances are measured to the designed focal nodes. We are rarely told where the nodes are designed, often both are inside the lens body somewhere, but some are outside. For telephoto lenses, the rear node H' (focal length from sensor plane) is just outside the front lens element, and its field H node is approximately the focal length dimension in front of the front lens surface (see example). The designer's term "telephoto" is about this reposition of the nodal point so that the physical lens is NOT longer than its focal length. And in wide angle lenses, the rear node H' is generally behind the rear lens surface (lens is moved well forward to provide room to allow the larger SLR mirror to rise... 12 mm lens but 24 mm mirror, etc). This nodal difference is only a few inches, but it affects where the focused distance is measured to the sensor. And it shifts a bit as the lens is focused much closer. Repeating, the focal length marked on the lens is specified for focus at infinity.

The Subject distance S is measured to the sensor focal plane (it is the "focus distance"), where we see a small line symbol like ϴ marked on the top of the camera (often near top LCD on right side, but some are on left side of camera). The line across the circle indicates the location of the film or sensor plane (for focus measurements). However, the Thin Lens Equation uses the working distance d in front of the lens. This is why we often see in equations: (S - f) used for d.


Again, three points.


For Macro, computing Field size from magnification is more convenient than from focal length, since we don't really know focal length at macro extension.

For example, we do know the Thin Lens Formula specifies focal length at 1:1 is extended to 2x the marked focal length at infinity. So FWIW, for the calculator Option 8 for magnification 1 (1:1) for a 50 mm lens at 2x, we could enter it as 100 mm, and compute field width for a full frame sensor as 0.11811 feet (which x12 and x25.4 is the expected 36 mm full frame width for 1:1). Subject distance becomes 0.32808 feet (both are feet if we use feet), which x12 is 3.9 inches. But this distance is measured to the Principle Point typically inside the lens somewhere, which we really don't know where that is, and it becomes very significant for macro. Which is why field of view for macro is instead computed from magnification (reproduction ratio, like 1:1).

Focal length and subject distance determine Magnification, which is the ratio of size of image to size of actual subject. Or the ratio of size of sensor to the size of the remote field. We could compute for magnification here, but we likely don't know new focal length at that close magnification. Just using magnification has more significance up closer (easier for macro), which is where our knowledge of the actual focal length is weakest. We could measure the field to compute the actual magnification. However Magnification is simply:

    m = s2/s1.   Or m = f/d.   Or m = f/(S-f).

Other macro ideas:

Actual focal length = Sensor dimension × Working distance / Field dimension

Those are all in the same units. Note that working dimension is to the lenses internal node, and NOT just to the front of the lens. Which is probably only an inch or two difference, but it becomes very significant at macro distances.

Field dimension = Sensor dimension / Magnification

Let's say it this way:

1:1 macro, the focal length f is same as the distance d in front of lens (each with its own node).

1:1 macro (magnification 1), the field of view is exactly the same size as the sensor.
1:2 macro (magnification 0.5), the field of view is twice the size of the sensor.
1:4 macro (magnification 0.25), the field of view is four times the size of the sensor.

This is true of any focal length for any lens (or method) that can achieve the magnification. Focal length and subject distance are obviously the factors determining magnification (it is still about them), but magnification ratio is simply easier work for macro.

The easiest method to determine field of view for macro is to simply put a mm ruler in the field. If a 24 mm sensor width sees 32 mm of ruler, then that is the field of view, and the magnification is 24/32 = 0.75 (this scale of magnification is 1 at 1:1, and is 0 at infinity).

The definition of macro 1:1 magnification is that the focal length and subject distance are equal (distances in front of and behind the lens nodes are necessarily equal, creating 1:1 magnification). In this Thin Lens Equation, if s1 and s2 are equal, the formula is then 2/s1 = 1/f, or 2f = s1. So lens extension to 2f gives 1:1. And since f/stop number = f / diameter, then if 2f, then f/stop number is 2x too, which a double f/stop number is 2 stops change, which is the aperture loss at 1:1. We know those things, this is just why.

But the point here, if f is actually 2f at 1:1 macro, the field of view changes with it. None of the FOV calculators are for macro situations (too close, magnification is instead the rule there). Field of View calculators expect subject distance to be at least a meter or so, reducing the focal length error to be insignificant.

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