A virtual scene has been illuminated with a point source located at a distance 2 m from the origin. Illumination at point (0, 0, 0) is given as I = I1rˆ + I0. Here, ˆr is unit vector along direction (1, 1, 1). Find out illumination at a point (10, 10, 10).
Question
A virtual scene has been illuminated with a point source located at a distance 2 m from the origin. Illumination at point (0, 0, 0) is given as I = I1rˆ + I0. Here, ˆr is unit vector along direction (1, 1, 1). Find out illumination at a point (10, 10, 10).
Solution
The given illumination at the origin (0,0,0) is I = I1rˆ + I0, where rˆ is the unit vector along the direction (1,1,1).
The unit vector rˆ is given by the direction vector divided by its magnitude. The direction vector from the origin to the point (1,1,1) is simply (1,1,1). The magnitude of this vector is sqrt(1^2 + 1^2 + 1^2) = sqrt(3). So, the unit vector rˆ = (1/sqrt(3), 1/sqrt(3), 1/sqrt(3)).
The illumination at a point (x,y,z) due to a point source is inversely proportional to the square of the distance from the source. So, if the source is at a distance 2m from the origin, the illumination at the origin is I1/(2^2) = I1/4.
Now, we want to find the illumination at the point (10,10,10). The distance from the origin to this point is sqrt(10^2 + 10^2 + 10^2) = sqrt(300). So, the illumination at this point is I1/(sqrt(300)^2) = I1/300.
Therefore, the illumination at the point (10,10,10) is I = I1/300 + I0.
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