I have $3$ points $(x_1,y_1,z_1)$, $(x_2,y_2,z_2)$, $(x_3,y_3,z_3)$ of the $3D$ plane and light source $(x_s,y_s,z_s)$.
Now, how can I find out the $N$ vector and $L$ vector?
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$\begingroup$ where are these three points? $\endgroup$ – janmarqz Mar 19 '18 at 16:56
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$\begingroup$ @janmarqz these are in original object plane. $\endgroup$ – alhelal Mar 19 '18 at 17:02
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Suppose, $P_1(x_1,y_1,z_1)$, $P_2(x_2,y_2,z_2)$, $P_3(x_3,y_3,z_3)$ and light source $S(x_s,y_s,z_s)$ are give. Then, $N$ is normal vector of cross product of $(P_2-P1)$ and $(P_3-P_1)$.
$(P_2-P1)=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}$
$(P_3-P_1)=(x_3-x_1)\hat{i}+(y_3-y_1)\hat{j}+(z_3-z_1)\hat{k}$
$\begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix}=N'$
$ \hat{i} \{(y_2-y_1)(z_3-z_1)-(y_3-y_1)(z_2-z_1)\}-\hat{j}\{ (x_2-x_1)(z_3-z_1)-(x_3-x_1)(z_2-z_1)\}+\hat{k}\{(x_2-x_1)(y_3-y_1)-(x_3-x_1)(y_2-y_1)\}=N'$
$ \hat{i} \{(y_2-y_1)(z_3-z_1)-(y_3-y_1)(z_2-z_1)\}+\hat{j}\{ (x_3-x_1)(z_2-z_1)-(x_2-x_1)(z_3-z_1)\}+\hat{k}\{(x_2-x_1)(y_3-y_1)-(x_3-x_1)(y_2-y_1)\}=N'$
$N=\frac{N'}{||N'||}$
Now, if $P_n(x,y,z)$ is a new point on the given plane then,(If you assume that brightness of each pixel of the polygon is same then you can consider this point as center of the polygon)
$L'=(S-P_n)$
$L'=(x_s-x)\hat{i}+(y_s-y)\hat{j}+(z_s-z)\hat{k}$
$L=\frac{L'}{||L'||}$
$N.L=N_xL_x+N_yL_y+N_zL_z$
$N.L=\{(y_2-y_1)(z_3-z_1)-(y_3-y_1)(z_2-z_1)\}(x_s-x)+\{(x_3-x_1)(z_2-z_1)-(x_2-x_1)(z_3-z_1)\}(y_s-y)+\{(x_2-x_1)(y_3-y_1)-(x_3-x_1)(y_2-y_1)\}(z_s-z)$
$N.L$ is needed for shading/brightness of a pixel(e.g. glcolor3f(N.L, N.L, N.L))
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1$\begingroup$ Great answer. I'd suggest one small change: often the vector from the surface to the light source is expected to be a unit vector, so I'd replace your $L$ with $L/\|L\|$. Also: sometimes light sources are "at infinity", and simply specified by the direction in which the light travels, i.e., some (unit) vector $v$; in that case, $L = -v$. $\endgroup$ – John Hughes Mar 20 '18 at 12:04
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$\begingroup$ Also, in the second to last line of equations, you have a $1/\|N\|$, which is unnecessary because you already normalized $N$. I think perhaps you should just simplify the expansion and says\ $N \cdot L = N_x L_x + N_y L_y + N_z L_z$, which then refers to the already-normallized coordinates. $\endgroup$ – John Hughes Mar 20 '18 at 17:10
