The Intermediate Value Theorem in Photography

黑墨墨

Photographers like to take photos at the extremes of the day(the half an
hour before sunrise or after sunset) at which time the blue end of the spectrum
would be scattered away, leaving the orange-red end of the spectrum to pen
-etrate the atmosphere. The atmosphere, serving as a gigantic reflector, also
gives dramatic lighting.

The challenge of taking photos at this time is waiting for the proper moment.
That is when the intermediate value theorem assumes its role in photography.
Take this photo of the Verrazano Narrows Bridge at sunset as an example.
The brightness of the sky rapidly decreases as the sun moves further down the
horizon, while the bridge towers are lit by the constant mercury lamps. (We
can ignore the reflected light from the atmosphere since the sun is behind the
bridge towers.) Mathematically, the brightness of the sky can be represented
as a continuous decreasing function in terms of time, f(t), while the brightness
of the towers is represented by a constant, B0. Take two different times, t1 and
t2, the brightness of the sky is f(t1) = B1 and f(t2) = B2. If B1 is too bright,
the details of the towers would not show up. If B2 is too dark, the details of
the sky would be lost. However, if B0 is between B1 and B2, the intermediate
value theorem guarantees a time t0 such that f(t0) = B0. At this time, the
brightness of the sky and the towers are the same, the details of the bridge and
the sky are preserved.

Many photographers might have taken great photos at the extremes of the
day without knowing the intermediate value theorem while many mathemati-
cians know the intermediate value theorem without knowing how to take great
photos.

nightfly

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