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photomacrographs Approach and normal lens APS-C Format and Full Format

Fact 1:
According to the figure below, the same that appeared in the article on the viewing angle, we can write the following formulas:
as α = m/2f
where:
α is half the solid angle.
m is the diameter of the circle circumscribing the sensor, the same for both the diagonal of the sensor.
f is the lens focal length.
If we know the value of F from the other two parameters, we find that the correct formula is:
; f = m / (2tan α)
Fact 2:
has been determined by empirical methods the angle of vision of the human being is, on average, 45 degrees. Or what is the same PI / 4 radians, and the tangent of the angle is well known by everyone. The unit. So 1.
Fact 3:
By trigonometry, the following formula gives the so-called double-angle tangent:
, so (2α) = (2tan α) / (1 - so ² α)
Since we know that tan (2α) = 1, if we want to know what the value of tan α, the sensible thing is to make the substitution: x = tan α
thus the above formula is reduced to us:
, 1 = (2x) / (1 - x ²)
which can be written as the following quadratic equation.
, x ² + 2x -1 = 0.
The solutions are, with two significant figures:
x1 = x2 = 0.41 and -2.41
Taking the positive solution, as in the physical world we are only in this case interesandos values absolute, we find that as α = 0.41
From this we can establish with relative ease which is the focal length "normal" an objective, defining this as providing a viewing angle similar to that of being human. Given that the formula appears as a parameter m, which is the diagonal of the sensor, it seems clear that the fact of having our camera with a sensor of APS-C or of a full format will provide different results. 's Check.
Complete Format sensor
In this case, the diameter of the sensor is 43 mm. Therefore:
f = 43 / (2 0.41) = 52.4
sensor APS-C
In this case, the diameter of the sensor is 30 mm. Therefore:
f = 30 / (2 0.41) = 36.5
For this reason it is not surprising that a target of 50 mm is considered normal when using a full format and 35mm when used APS-C format. It is clear that manufacturers of goals would not use decimal numbers in them. The only thing missing.