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Understanding Vector Reflection Visually September 23, 2012

Posted by Andor Saga in Game Development, Math.

I started experimenting with normal maps when I came to the subject of specular reflection. I quickly realized I didn’t understand how the vector reflection part of the algorithm worked. It prompted me to investigate exactly how all this magic was happening. Research online didn’t prove very helpful. Forums are littered with individuals throwing around the vector reflection formula with no explanation whatsoever. This forced me to step through the formula piecemeal until I could make sense of it. This blog post attempts to guide readers through how one vector can be reflected onto another via logic and geometry.

Vector reflection is used in many graphics and gaming applications. As I mentioned, it is an important part of normal mapping since it is used to calculate specular highlights. There are other applications other than lighting, but let’s use the lighting problem for illustration.

Let’s start with two vectors. We would know the normal of the plane we are reflecting off of along with a vector pointing to the light source. Both of these vectors are normalized.

L is a normalized vector pointing to our light source. N is our normal vector.

We are going to work in 2D, but the principle works in 3D as well. What we are trying to do is figure out: if a light in the direction L hits a surface with normal N, what would be the reflected vector? Keep in mind, all the vectors here are normalized.

I drew R here geometrically, but don’t assume we can simply negate the x component of L to get R. If the normal vector was not pointing directly up, it would not be that easy. Also, this diagram assumes we have a mirror-like reflecting surface. That is, the angle between N and L is equal to the angle between N and R. I drew R as a unit vector since we only really care about its direction.

So, right now, we have no idea how to get R.
R = ?

However, we can use some logic to figure it out. There are two vectors we can play with. If you look at N, you can see that if we scale it, we can create a meaningful vector, call it sN. Then we can add another vector from sN to R.

What we are doing here is exploiting the parallelogram law of vectors. Our parallelogram runs from the origin to L to sN to R back to the origin. What is interesting is that this new vector from sN to R has the opposite direction of L. It is -L!

Now our formula says that if we scale N by some amount s and subtract L, we get R.
R = sN - L

Okay, now we need to figure out how much to scale N. To do this, we need to introduce yet another vector from the origin to the center of the parallelogram.

Let’s call this C. If we multiply C by 2, we can get sN (since C is half of the diagonal of the parallelogram).
2C = sN

Replacing sN in our formula with 2C:
R = 2C - L

Figuring out C isn’t difficult, since we can use vector projection. If you are unsure about how vector projection works, watch this KhanAcademy video.

If we replace C with our projection work, the formula starts to look like something! It says we need to project L onto N (to get C), scale it by two then subtract L to get R!
R = 2((\frac{N \cdot L}{|N|^2})N) - L

However, this can be simplified since N is normalized and getting the magnitude of a normalized vector yields 1.
R = 2(N \cdot L)N - L

Woohoo! We just figured out vector reflection geometrically, fun!

I hope this has been of some help to anyone trying to figure out where the vector reflection formula comes from. It has been frustrating piecing it all together and challenging trying to explain it. Let me know if this post gave you any ‘Ah-ha’ moments (: