I have a 3D graphics book, which gives the formula for absorption of radiance along a ray. I am trying to derive the details and would like to see if my derivation is correct.(adsbygoogle = window.adsbygoogle || []).push({});

Let o(p) be the probability density that light is absorbed per unit length at point p.

They give the formula as: exp( -int_0^d(o(p+tv)dt) )

where p is the starting point of the ray entering the medium, and it exits at the point p+dv.

So to set this up, I look at a small section along the ray, to see how much light is absorbed across that small section of length h. I also recast things in terms of t, since the position along the ray is a function of t. Let L(t) denote the light radiance at point t.

L(t+h) = L(t) - o(t)*h*L(t)

That is, the light radiance after passing through a segment of length h equals the incoming light radiance minus the amount absorbed.

[L(t+h) - L(t)] / h + o(t)*L(t) = 0

Taking the limit h-->0

dL(t) / dt + o(t)*L(t) = 0

I multiply by integrating factor exp(int_0^t(o(t)dt))

to get

(exp(int_0^t(o(t)dt)) * L(t))' = 0

Integrating from 0 to d:

exp(int_0^d(o(t)dt))*L(d) - exp(int_0^0(o(t)dt))*L(0) = C

We know L(0) = 0 since no absorption yet.

L(d) = C*exp(-int_0^d(o(t)dt))

I'm not sure how to get rid of the C. I think it should not have occurred since I use definite integral. Also, is my integrating factor correct: exp(int_0^t(o(t)dt))??

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# Total absorption through medium along ray

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