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Let $I(x,y)$ be the intensity at position $(x,y)$, $\rho(x,y)$ be the albedo at this point (a constant), $n=(p,q,1)$ be the surface gradient vector and $s_{0},s_{1},s_{2}$ all be vectors from the point to $(x,y)$.

Assume that $I(x,y)=I_{1}$ when $s_{0}=(1,1,0)$ and similarly, $I(x,y)=I_{2}$ when $s_{1}=(1,0,1)$ and $I(x,y)$ = $I_{3}$ when $s_{2}=(0,1,0)$

Lambert's cosine Law is then:

$I(x,y)=\rho(x,y)\dfrac{s_{x}p+s_{y}q+s_{z}}{|n||s|}$

We can calculate the following. $|n|=\sqrt{p^{2}+q^{2}+1}$ and $|s_{0}|=|s_{1}|=|s_{2}|=1$.

All I would like to have are equations for $p$ and $q$ in terms of $I_{1},I_{2},I_{3}$.

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So you know the values of $s_xp+s_yq+s_z$ for three linearly independent vectors $s$. Linear algebra does the rest. –  darij grinberg Mar 31 '11 at 21:17
    
I don't understand how $s_0=(1,1,0)$ is compatible with $|s_0|=1$. –  Gerry Myerson Mar 31 '11 at 23:38
    
darij: Not quite. He knows your quantities divided by a function of p and q. He needs a little more than linear algebra to find p and q. Gerhard "Ask Me About System Design" Paseman, 2011.03.31 –  Gerhard Paseman Apr 1 '11 at 1:18
    
The way the question is phrased, I actually cannot make out the hypotheses. What does "vectors from the point to (x,y)" mean? Why do the specified s_0 and s_1 have length 1? –  Will Jagy Apr 1 '11 at 4:20
1  
You might write out what you think are the three equations giving I1, I2, and I3 in terms of p and q, filling in as much of the known values as you can. We might then tell you where your arithmetic is wrong, or we might come up with the answer for you. In any case, this is looking more like something for math.stackexchange than for MathOverflow. Gerhard "Ask Me About System Design" Paseman, 2011.04.01 –  Gerhard Paseman Apr 1 '11 at 9:20
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closed as too localized by Will Jagy, darij grinberg, Gerry Myerson, Franz Lemmermeyer, Andy Putman Apr 1 '11 at 16:08

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