C++ raytracer and normalizing vectors

Question

So far my raytracer:

1. Sends out a ray and returns a new vector if collision with sphere was made

2. Pixel color is then added based on the color of the sphere[id] it collided with.

3. repeats for all spheres in scene description.

For this example, lets say:

sphere[0] = Light source
sphere[1] = My actual sphere

So now, inside my nested resolution for loops, I have a returned vector that gives me the xyz coordinates of the current ray's collision with sphere[1].

I now want to send a new ray from this collision vector position to the vector position of the light source sphere[0] so I can update the pixel's color based off this light's color / emission.

I have read that I should normalize the two points, and first check if they point in opposite directions. If so, don't worry about this calculation because it's in the light's shadow.

So my question is, given two un-normalized vectors, how can I detect if their normalized unit's are pointing in opposite directions? And with a point light like this, how could that works since each point on the light sphere has a different normal direction? This concept makes much more sense with a directional light.

Also, after I run this check, should I do my shading calculations based off the two normal angles in relationship to each other, or should I send out a new ray towards the lighsource and continue from there?

Answer 1

You can use the dot product of the 2 vectors, that would be negative if they are in the opposite direction, ie the projection of one vector onto another is going in the opposite direction

Answer 2

For question 1, I think you want the dot product between the vectors.

u.v = x1*x2 + y1*y2 + z1*z2

If u.v > 0 then the angle between them is acute.

if u.v < 0 then the angle between them is obtuse.

if 0.v == 0 they point at exactly 90 degree angle.

But what I think you really mean is not to normalize the vectors, but to compute the dot product between the angle of the normal of the surface of the sphere at your collision xyz to the angle from your light source to the same xyz.

So if the sphere has center at xs, ys, zs, and the light source is at xl, yl, zl, and the collision is at xyz then

vector 1 is x-xs, y-ys, z-zs and vector 2 is xl-x, yl-y, zl-z

if the dot product between these is < 0 then the light ray hit the opposite side of the sphere and can be discarded.

Once you know this light ray hit the sphere on the non-shadowed side, I think you need to do the same calculation for the eye point, depending on the location of the light source and the viewpoint. If the eye point and the light source are the same point, then the value of that dot product can be used in the shading calculation.

If the eye and light are at different positions the light could hit a point the eye can't see (and will be in shadow and thus ambient illumination if any), so you need to do the same vector calculation replacing the light source coordinate with the eye point coordinate, and once again if the dot product is < 0 it is visible.

Then, compute the shading based on the dot product of the vector from eye, to surface, and surface to light.

OK, someone else came along and edited the question while I was writing this, I hope the answer is still clear.