# Regarding the calculation of Focal length, angle of view, Sensor size from known working distance, smallest field of view of the sensor (vertical)

I´m looking for a high-image quality machine vision camera with suitable lenses. The specifications are a working distance of 600 mm, a Field of view of the smallest side of the sensor is 400 mm (vertical side) and a depth of field of 100 mm. I calculated the angle of view using working distance and field of view. That is using the triangular formula:

tan(AOV/2) = (FOV/(2*W.D:))

AOV: Angle of view of the lens FOV: field of view W.D.: Working Distance

I found out that the AOV is 36.87°. Is this correct? Then, how to calculate the focal length, sensor sizes and resolution based on these parameters? Please give some suggestions and guidance. I appreciate any help you can provide.

The image projected by the camera lens is circular. Only the central portion of this circular image is suitable for pictorial work. The unusable outer portion of this image is baffled away by the camera. The central usable portion is called the circle of good definition. As a rule-of-thumb we mount a lens with a focal length equal to or greater than the diagonal measure of the image sensor. This assures the lens chosen will provide reasonable definition and uniformity over the entire area of the sensor. If a lens is mounted that is shorter than diagonal of the imaging surface, it must be specially designed to cover this image area without vignetting.

The image triangle (ray trace) you have made is accurate. The object triangle you have not traced has its apex at the lens. The dimensions of this object triangle is proportional to the image triangle. All angles are the two triangles are identical. You have only provided the vertical dimension of the image rectangle. We need the horizontal dimension of the image rectangle to calculate the diagonal of image rectangle. Once the diagonal is known, I suggest the focal length to use is a lens focal length equal to or slightly longer than this value.

• Thank you for your great suggestion. I still have some questions. When selecting a horizontal field of view (FOV) of 450 mm and a vertical FOV of 250 mm, the diagonal FOV is calculated to be 514.78 mm. We can calculate the angle of view (AOV) using the following formula: tan(AOV/2) = (FOV/(2*W.D)) WD: 600 mm FOV: 514.78 mm The calculated angle of view on the sensor's longest dimension (diagonal) is 46.44°, horizontally is 21.24°, and vertically is 11.89°. Based on these values, how can I determine the focal length of the lens? Thank you very much for your kind support. Commented Aug 1 at 9:40
• Lens focal length same as diagonal will likely yield the uniformity you desire. That's a lens 514.8mm. Suppose you buy a 550mm. Magnification 1 achievable with any focal length lens. “Unity” set-up 2X focal length from image sensor to lens and 2X focal length lens to object to lens. Distance lens to sensor is 4X the focal length. This will give you good uniformity. For format is 250mm Vertical 450 horizontal 514.8 diagonal Angle of view 25.6° vertical 44.5° horizontal 50.2° diagonal Sensor to subject distance 2200mm Lens to subject distance 1100mm Lens to sensor distance 1100mm Commented Aug 1 at 14:49

I found out that the AOV is 36.87°. Is this correct?

Not quite. The formula is fine; however the lens' FOV required for a rectilinear sensor is equal to the sensor diagonal. If the sensor is a square format with 200mm sides then the required FOV needs to cover 283mm. I.e. the sensor's longest dimension (diagonal) must be ≤ the FOV diameter. So the FOV required is ~ 51˚. (you calculated the lens FOV diameter required to record 200mm short edge; except that the sensor's short edge/side won't be recorded from the FOV center (diameter)).

You can then use a lens FOV chart like this one (based on a 35mm sensor format).

And then use the sensor's crop factor to determine the required focal length ("crop factor" is also based on the 35mm format). E.g. a 4/3 sensor has a 2x crop factor, and therefore needs a lens with ~ 100˚ FOV to record 50˚ on the sensor... ~ 18mm.

But there are a lot of variables that can come into play... e.g a lot of times a lens' reported focal length is rounded somewhat, and it is often less when the lens is focused at distances short of infinity.

And focal length is sometimes reported as "35mm equivalent" which already accounts for the crop factor. I.e. for a sensor lens combination you would want one with ~ a "45mm equivalent" lens.

Then you can use the circle of confusion (CoC/airy disk size) requirement to determine the lens' aperture and sensor resolution for the plane of focus.

This chart is relevant:

However, that doesn't account for the required depth of field. That would depend on the required enlargement for viewing... i.e. if the sensor's physical size needs enlarged 100 times for viewing, then you need a CoC/airy disk 1/100 the size listed in the table, and a pixel size 1/2 of that (or 1/4 for maximum color resolution).

• That was a great suggestion from your side. As I´m new to this topic. Still I have some doubts. Commented Aug 1 at 9:23
• Thank you for your excellent suggestion. When selecting a horizontal field of view (FOV) of 450 mm and a vertical FOV of 250 mm, the diagonal FOV is calculated to be 514.78 mm. From this value, we can calculate the angle of view (AOV) using the following formula: tan(AOV/2) = (FOV/(2*W.D)) WD: 600 mm FOV: 514.78 mm The calculated angle of view on the sensor's longest dimension (diagonal) is 46.44°, horizontally is 21.24°, and vertically is 11.89°. Based on these values, how can I determine the focal length of the lens? Thank you very much for your kind support. Commented Aug 1 at 9:39
• You need the sensor's actual dimensions (area) in relation to a 35mm sensor 36x24 (43.3 diag)). E.g. an m4/3 sensor is 17.3x13 (21.64 diag)... the sensor diagonal is 1/2 of the value of the 35mm full frame camera, so it has a 2x crop factor and it will only record 1/2 of the lens' FOV. I.e to record 46.5˚ FOV you need a lens with a 93˚ FOV. Commented Aug 1 at 11:44
• Suppose I use a 35mm full-frame sensor; can a 35mm lens record a 46.5° FOV? Commented Aug 1 at 14:03
• From the chart above; a 50mm projects a 45˚ FOV on a full frame sensor, and a 35mm projects a 64˚ FOV. It's not in the chart, but a 45mm lens projects a 50˚ FOV; and it is probably the best choice for recording a 46.5˚ FOV with a full frame sensor. It's also good to know that the reported/designed FOV is the area meant to be recorded. The projected image circle is usually larger than that, but of lower quality at the edges. Commented Aug 1 at 14:46

The easiest way to do much of this is just with the focal length of the lens and pretending it's a pinhole.

The values you need are focal-length and sensor dimensions.

You take the focal length of the lens at the focus you're using (it shifts a little), and make it the point of a pyramid that distance from the sensor with the corners of the sensor as the other corners. You then project that pyramid past the point to get the view frustrum of the camera.

You can use this imaginary pyramid to calculate vertical and horizontal angle of view with a little trigonometry.

Trying to calculate any of this from the imaging circle is probably unreliable as camera designs can choose to not push so close to the edge of the circle to reduce vignetting.

• I don´t know the focal length and sensor dimension. In this case, I have only known parameters FOV: 250mm (vertical), FOV: 450mm (horizontal), FOV: 514.78 (diagonal) and a working distance of 600 mm. From this, I need to find out the focal length of the lens and select a camera sensor that suits this specification. This industrial camera with a good resolution for capturing different cloth materials. Commented yesterday
• @Ajeesh it sounds like maybe all of your figures are on the subject side of the lens. In that case there's no info for the sensor side and you can't calculate that. I think you need a better data sheet Commented yesterday