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My problem is the following: I am looking for all possible functions $f(\theta_\text{i}, \theta_\text{o})$ with $f(\theta_\text{i}, \theta_\text{o}) \in \mathbb{R} \ \forall \ (\theta_\text{i}, \theta_\text{o}) \in [0, \pi/2]^2$ that verify the following equation and conditions:

$$\int_0^{\pi/2} f(\theta_\text{i}, \theta_\text{o}) \sin 2\theta_\text{o} \, \text{d}\theta_\text{o} = \sin^2 \theta_\text{i}$$

$$f(\theta_\text{i}, \theta_\text{o}) = f(\theta_\text{o}, \theta_\text{i}) \quad \text{and} \quad f(\theta_\text{i},\theta_\text{o}) \geq 0 \ \forall \ (\theta_\text{i}, \theta_\text{o})$$

Even if the question is simple, I am struggling to start... I have tried integration by part, I also tried to derivate by $\theta_\text{i}$ in both sides, or the King's property... But I am going nowhere.

I will take any advice.

Balfar
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    Where did this problem come from? – Ted Shifrin Apr 18 '23 at 18:28
  • By integrating by parts, this can become: $$\sin^2(\theta_\text{i}) = f\left(\theta_\text{i}, \pi/2\right) + \int_0^{\pi/2} \frac{\partial f}{\partial \theta_\text{o}}(\theta_\text{i}, \theta_\text{o}) \sin^2(\theta_\text{o}) , \text{d}\theta_\text{o}$$ meaning that it's like an integral equation with a kernel except you'd solve for the kernel in a way? By that I mean it's of the form $u(x) - \int_a^b K(x,y) u(y) \mathrm{d} y = g(x)$ but you'd solve for $K$. – Bruno B Apr 18 '23 at 19:11
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    @TedShifrin I am studying physically based rendering, and this problem applies to a material reflectivity model that I am trying to develop. The special condition required corresponds to a special case of the Helmholtz reciprocity principle. – Balfar Apr 18 '23 at 19:12
  • Ah it should be a $-$ in my first equation but five minutes have passed so I cannot edit it. – Bruno B Apr 18 '23 at 19:19
  • This suggests that $-\partial f/\partial\theta_0$ should be a delta function concentrated on the diagonal. This is not compatible with the desired symmetry of $f$. – Ted Shifrin Apr 18 '23 at 21:11

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Split the LHS integral from 0 to $\theta_i$ then from $\theta_i$ to $\frac{\pi}{2}$ and you also have : $\sin^2(\theta_i) = \int_{0}^{\theta_i} \sin(2 \theta_o) d \theta_o$

HellBoy
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