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F̶i̶n̶d̶ ̶a̶l̶l̶ ̶a̶n̶g̶l̶e̶s̶ (edit: that might have been asking for too much, sorry for the wrong problem statement, I thought that by solving for all angles, finding x would be trivial but as it seems to be thanks to timon92 that might not be possible) Find the angle $x$ knowing only the length of the side |AB|, the constants $E_A$ and $E_B$ and the relation $$E_B r_B^2 sin(\gamma) = E_A r_A^2sin(\delta)$$ enter image description here enter image description here

I've tried to solve a system of equations containing the known relation the sine law, the three cosine laws for each of the sides along with the pythagorean identities. enter image description here enter image description here I wasn't even sure if this had a solution but I've managed to reduce it to a system of 4th degree polynomials and I'm still not sure if this is doable. Any sort of help would be greatly appreciated. (Edit: the second to last bracket is wrong but that doesn't change much, it's still a not an easy system of equations)

The give a bit of context the problem comes from my own project, where there is a light point source at C, two luminosity readings at A and B, $E_i = \frac{I}{r_i^2}cos(\theta) $, where I is the unknown intensity and $\theta$ is the angle between the illuminated surface and the plane perpendicular to the incident light. Originally I wanted to find the azimuth and altitude angles (x in the 2d case) of the light source given three sensors and their known relative position without the distance (so 2 out of 3 spherical coordinates) but even the 2d case stumped me.

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2 Answers 2

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The problem is underconstrained. There are several triangles satisfying this condition.

By the sine law we have $\dfrac{r_A}{\sin \delta} = \dfrac{r_B}{\sin \gamma}$ , hence the condition is equivalent to $$\frac{r_A}{r_B} = \left(\frac{E_B}{E_A}\right)^{1/3}.$$ The locus of points $C$ satisfying this equation is a certain circle (or a line if $E_A=E_B$) called Apollonius circle.

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  • $\begingroup$ What about the knowledge of the side length of |AB| and the cosine laws, don't they contribute enough to ensure a unique solution? $\endgroup$
    – user379685
    Commented May 7, 2021 at 11:22
  • $\begingroup$ Ok I can see what you mean. I might have asked for too much by saying to find all the angles. Is there a way to find the angle x at which the point C is tilted in relation to the side |AB|? $\endgroup$
    – user379685
    Commented May 7, 2021 at 11:52
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    $\begingroup$ @user379685 No, there is not. $x$ is not determined uniquely. $\endgroup$
    – timon92
    Commented May 7, 2021 at 13:29
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HINTS:

The polar relation

$$( r/a)^2= \sin \theta , \text{ which wlog can be written as :} $$

$$( r_A= \sqrt{\sin \gamma \;E_B},\; r_B= \sqrt{\sin \delta \;E_A} ) $$

represent a drop / flattened circle shapes of maximum heights as shown.

We can find intersections of these flattened circles... like graphically shown in the following or numerically from analytical form:

$$r_A \cos \theta_A-r_B \cos \theta_B= AB,\; r_A \sin \theta_A=r_B \sin \theta_B;\;$$

$$ \theta_A = \gamma,\; \theta_B= \pi- \delta\;$$

enter image description here

The construction shows heights $ (1,1.5) $ of drop shapes cutting at two points and corresponding four polar angles on common base $( AB=1.25)$

Asking for all solutions is okay.. we should see what is relevant to the Physics and discard what is not.

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