Comment by keyle

I used to feel like you before I went to university and had a few math courses. Then it became a lot more clear.

And it really isn't that bad in most cases, and isn't unlike how we learnt that "int 10h" is how you change graphics modes[1] in DOS back in the days.

The "big S" is an integral, which is in most cases essentially a for-loop but for continuous values rather than discrete values. You integrate over a domain, like how you can for-loop over a range or discrete collection.

The domain is written here as just a collection of continuous values S^2, so like a for-in loop, though it can also be from and to specific values in which case the lower bound is written subscript and the upper bound superscript.

Similar to how you have a loop variable in a for-loop you need an integration variable. Due to reasons this is written with a small "d" in front, so in this case "dω_i" means we're integration (looping) over ω_i. It's customary to write it either immediately after the integral sign or at the end of the thing you're integrating over (the loop body).

However dω_i serves a purpose, as unlike a regular discrete for-loop, integrals can be, lets say, uneven, and the "d" term serves to compensate for that.

The only other special thing is the use of the absolute function, written as |cosθ_i|, which returns the absolute value of cosθ_i, the cosine of the angle θ_i. Here θ_i is defined earlier in the book as the vertical angle of ω_i relative to the surface normal at the point in question, which can be calculated using the dot product.

So in programmer-like terms, it reads a bit like this in pseudo-code.

    function L_o(p, ω_o): 
      integral_value = 0
      for ω_i in S^2 do
        θ_i = dot(w_i, n)
        integral_value += f(p, ω_o, ω_i) * L_i(p, ω_i) * abs(cos(θ_i)) * dω_i
      return L_e(p, ω_o) + integral_value

Note that the surface normal "n" is implicitly used here, typically it would be passed explicitly in code.

What's special here is that unlike a normal for-loop, in math the changes in ω_i, represented by dω_i, are infinitesimally small. But in practice you can actually implement a lot of integrals by assuming the changes are small but finite[2].

Anyway, this wasn't meant as a full explanation of integrals and such, but just an attempt to show that it's not all gobbledygook.

[1]: https://en.wikipedia.org/wiki/INT_10H

[2]: https://en.wikipedia.org/wiki/Riemann_sum