Another 35mm lens only has Field of View specified for FF (35mm) 63 degrees, can I mathematically find what FOV this lens will have on APS-C (crop factor 1.5) sensor?
Assuming the lens is rectilinear, which is a safe assumption for almost all lenses that aren't fisheye, then you can use the pinhole projection mapping function to calculate angle of view:
As Alan Marcus mentioned in his answer, the listed angle of view of most lenses is with respect to the camera's diagonal film/sensor plane. 35mm full-frame camera imaging planes are 36mm wide by 24mm tall, meaning the diagonal dimension is 43.3 mm.
Most non-Canon APS-C lenses are said to use a crop factor of roughly 1.5, meaning the sensor is 1.5 times smaller in height, width, and diagonal dimension than full-frame sensors. This is close, but in reality, general APS-C sensor heights are 15.6 mm high, meaning compared to full-frame's 24 mm sensor height, they are 24 / 15.6 = 1.54 times smaller.
(Canon uses an APS-C crop factor of 1.6, meaning their sensors are just slightly smaller than APS-C sensors from Nikon, Sony, etc.)
So for the ƒ = 35 mm lens you are asking about, using d = 43.3 / 1.54 = 28.1 mm, the angle of view is ⍺ = 44º on a non-Canon APS-C camera body.
The pinhole projection mapping function just means that light rays entering the lens system, that are aimed towards the optical center, exit the lens system from the optical center at the same angle, just as if the lens were replaced with an arbitrarily small pinhole:
This camera diagram shows light entering a camera's lens from the left, into a lens of focal length ƒ and an ange of view ⍺, projected onto a sensor of size (dimension) d (it doesn't matter if the dimension is height, width, or diagonal; as long as you're consistent, the formula works for any of them).
The pinhole formula is just the solution of the red triangle with angle "⍺/2", with a cosine length of ƒ (the "run"), and a sine length of "d/2" (the "rise"). The tangent of the angle is just the sine divided by the cosine, the "rise over the run".
Double checking the math on the 20mm lens you cited:
- plugging in d = 43.3 mm, ƒ = 20 mm, the angle of view ⍺ = 94.5º on a full frame body.
- plugging in d = 43.3 / 1.54 = 28.1 mm, ƒ = 20 mm, the angle of view ⍺ = 70.2º on a non-Canon APS-C camera body.
These are very close to the specs listed. That verifies that the lens uses a pinhole projection mapping function.