Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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:


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 $I_{1},I_{2},I_{3}$.

share|improve this question
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
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
show 5 more comments

closed as too localized by Will Jagy, darij grinberg, Gerry Myerson, Franz Lemmermeyer, Andy Putman Apr 1 '11 at 16:08

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.