Note that, if you have the sensor size and horizontal or vertical fov's, you can solve one of the first two equations for f and plug it into the third one to get the diagonal fov.

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Tube lens

When, as is usual, the focal length is estimated through camera calibration, and is expressed in pixels, the above expressions need some adapting.

In the simplest case the focal axis is orthogonal to the image plane (K12 = 0), the pixels are square (K11 = K22), and the principal point is at the image center (K13 = Wp/2; K23 = Hp/2). Then the same equations as above apply, replacing W with Wp, H with Hp and f with K11.

Liquidlens edmund optics

Denote with K the 3x3 camera matrix, with the camera frame having its origin at the camera center (focal point), X axis oriented left-to-right, Y axis top-to-bottom and Z axis toward the scene. Let Wp and Hp respectively be the width and height of the image in pixels.

Liquidcrystal lens

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But liquid lenses are more flexible, allowing them to shift between all of these focal lengths easily. Liquid lenses themselves are nothing new – tech companies have been promising their arrival in consumer electronics since at least 2004 – but the first one only made it to market last year, in Xiaomi’s Mi Mix phone.

Deep optics

In tests, the liquid lens was able to change its focal length from 7.5 mm to 13.1 mm, as the voltage was increased from zero to 100 V. Its resolution was almost 29 lines per millimeter, and it could allow about 95 percent of visible light through. The liquid lens also performed well in a range of temperatures, between 20 and 100 °C (68 and 212 °F).

Existing lenses are solid, and while they work well, they have a limited range of focal lengths. That’s why modern smartphones are packing in two, three, four or more lenses lately, specialized for macro, wide-angle, or telephoto shots.

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The team says that the new liquid lens has a simpler structure and better stability than existing liquid lenses. Some require insulating layers between the electrode and the liquid, while others such as Xiaomi’s use tiny motors to manipulate their shape. Without those, the DBA lens should be easier to integrate into smartphones, cameras, endoscopes, and other lenses.

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A lil more complex is the case just as above, but with the principal point off-center. Then one simply adds the two sides of each FOV angle. So, for example:

Tunable lens

The new liquid lens, developed by researchers at Hefei University of Technology, is designed to be simpler than existing tech. The key ingredient is a fluid known as dibutyl adipate (DBA), which has an electronegative molecular structure. That means that its shape will change when a direct current is applied to it.

Is there a mathematical relation between those values? If I know hFOV and vFOV can I calculate the diagonal FOV without involving other values like focal lengths etc?

The lens was created by filling a ring-shaped electrode with this DBA liquid. Because the electrode is coated with a water-repelling layer, the surface of the liquid forms a dome shape. When a voltage is applied to it, the DBA molecules gather at one end of the tube, changing the dome and, as a result, the focal length of the lens. When the electric field is removed, the liquid returns to its original shape.

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Researchers in China have developed a new type of adaptive liquid lens that changes its focal distance when a voltage is applied. Made using a new electrically responsive fluid, the lens is small, lightweight, and could be easily manufactured for use in smartphones and other cameras.

If the pixels are not square the same expressions apply for FOV_vertical, but using K22 and Hp, etc. The diagonal is a tad trickier, since you need to "convert" the image height into the same units as the width. Use the "pixel aspect ratio" PAR=K22/K11 for this purpose, so that:

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The physical quantities of interest are the sensor size and the focal length. The latter, in the pinhole camera model, is the the distance between the camera center and the image plane. Therefore, if you denote with f the focal length (in mm), W and H respectively the image sensor width and height (in mm), and assume the focal axis is orthogonal to the image plane, by simple trigonometry it is: