RayTracing: When to Normalize a vector?

Question

I am rewriting my ray tracer and just trying to better understand certain aspects of it.

I seem to have down pat the issue regarding normals and how you should multiply them by the inverse of the transpose of a transformation matrix.

What I'm confused about is when I should be normalizing my direction vectors?

I'm following a certain book and sometimes it'll explicitly state to Normalize my vector and other cases it doesn't and I find out that I needed to.

Normalized vector is in the same direction with just unit length 1? So I'm unclear when it is necessary?

Thanks

Answer 1

You never need to normalize a vector unless you are working with the angles between vectors, or unless you are rotating a vector.

That's it.

In the former case, all of your trig functions require your vectors to land on a unit circle, which means the vectors are normalized. In the latter case, you are dividing out the magnitude, rotating the vector, making sure it stays a unit, and then multiplying the magnitude back in. Normalization just goes with the territory.

• If someone tells you that coordinate system are defined by n unit vectors, know that i-hat, j-hat, k-hat, and so on can be any arbitrary vector(s) of any length and direction, so long as none of them are parallel. This is the heart of affine transformations.

• If someone tries to tell you that the dot product requires normalized vectors, shake your head and smile. The dot product only needs normalized vectors when you are using it to get the angle between two vectors.

But doesn't normalization make the math "simpler"?

Not really -- It adds a magnitude computation and a division. Numbers between 0..1 are no different than numbers between 0..x.

Having said that, you sometimes normalize in order to play well with others. But if you find yourself normalizing vectors as a matter of principle before calling methods, consider using a flag attached to the vector to save yourself a step. Mathematically, it is unimportant, but practically, it can make a huge difference in performance.

So again... it's all about rotating a vector or measuring its angle against another vector. If you aren't doing that, don't waste cycles.

Answer 2

tl;dr: Normalized vectors simplify your math. They also reduce the number of very hard to diagnose visual artifacts in your images.

Normalized vector is in the same direction with just unit length 1? So I'm unclear when it is necessary?

You almost always want all vectors in a ray tracer to be normalized.

The simplest example is that of the intersection test: where does a bouncing ray hit another object.

Consider a ray where:

p(t) = p_0 + v * t

In this case, a point anywhere along that ray p(t) is defined as an offset from the original point p_0 and an offset along a particular direction v. For every increment of parameter t, the resulting p(t) will move another increment of length equal to the length of the vector v.

Remember, you know p_0 and v. When you are trying to find the point where this ray next hits another object, you have to solve for that t. It is obviously more convenient, if not always obviously necessary, to use normalized vector vs in that representation.

However, that same vector v is used in lighting calculations. Imagine that we have another direction vector u that points towards a lighting source. For the purpose of a very simple shading model, we can define the light at a particular point to be the dot product between those two vectors:

L(p) = v * u

Admittedly, this is a very uninteresting reflection model but it captures the high points of the discussion. A spot on a surface is bright if reflection points towards the light and dim if not.

Now, remember that another way of writing this dot product is the product of the magnitudes of the vectors times the cosine of the angle between them:

L(p) = ||v|| ||u|| cos(theta)

If u and v are of unit length (normalized), then the equation will evaluate to be proportional to the angle between the two vectors. However, if v is not of unit length, say because you didn't bother to normalize after reflecting the vector in the ray model above, now your lighting model has a problem. Spots on the surface using a larger v will be much brighter than spots that do not.

Answer 3

It is necessary to normalize a direction vector whenever you use it in some math that is influenced by its length.

The prime example is the dot product, which is used in most lighting equations. You also sometimes need to normalize vectors that you use in lighting calculations, even if you believe that they are normal.

For example, when using an interpolated normal on a triangle. Common sense tells you that since the normals at the vertices are normal, the vectors you get by interpolating are too. So much for common sense... the truth is that they will be shorter unless they incidentially all point into the same direction. Which means that you will shade the triangle too dark (to make matters worse, the effect is more pronounced the closer the light source gets to the surface, which is a... very funny result).

Another example where a vector might or might not be normalized is the cross product, depending on what you are doing. For example, when using the two cross products to build an orthonormal base, then you must at least normalize once (though if you do it naively, you end up doing it more often).
If you only care about the direction of the resulting "up vector", or about the sign, you don't need to normalize.

Answer 4

I'll answer the opposite question. When do you NOT need to normalize? Almost all calculations related to lighting require unit vectors - the dot product then gives you the cosine of the angle between vectors which is really useful. Some equations can still cope but become more complex (essentially doing the normalization in the equation) That leaves mostly intersection tests.

Equations for many intersection tests can be simplified if you have unit vectors. Some do not require it - for example if you have a plane equation (with a unit normal) you can find the ray-plane intersection without normalizing the ray direction vector. The distance will be in terms of the ray direction vectors length. This might be OK if all you want is to intersect a bunch of those planes (the relative distances will all be correct). But as soon as you want to compare with a different distance - calculated using the normalized ray direction - the distance values will not compare properly.

You might think about normalizing a direction vector AFTER doing some work that does not require it - maybe you have an acceleration structure that can be traversed without a normalized vector. But that isn't relevant either because eventually the ray will hit something and you're going to want to do a lighting/shading calculation with it. So you may as well normalize them from the start...

In other words, any specific calculation may not require a normalized direction vector, but a given direction vector will almost certainly need to be normalized at some point in the process.