# Raytracing Reflection, Refraction, Fresnel, Total Internal Reflection, and Beer’s Law

This post talks about how to render images like the below in real time using ray tracing. Some realism in the images come from reflection and refraction, but the real icing on the cake comes from Fresnel, total internal reflection and Beer’s law. We’ll look at the contributions of these features individually and talk about how to properly combine them for the greatest and most realistic results (:

The renderings come from a shadertoy that accompanies this post: Shadertoy: Reflect Refract TIR Fresnel RayT

My motivation for learning more about this stuff is that I’m starting to make a marble madness inspired game, and I’m planning to do hybrid rendering between rasterization and ray based techniques.

This post will focus more on how to make these features work for you in ray tracing, and less about the reasons for why they work the way they do. This post is more about practical implementations, and less about rigorous explanations.

If you have any questions, comments, corrections or additions, feel free to leave in the comments section at the bottom, or feel free to hit me up on twitter at @Atrix256.

This post is assumes you know how to do basic raytracing with ambient, diffuse and specular lighting, like the image below, which we are going to start with:

# Reflection

The first thing to talk about is reflection. More specifically we are going to be talking about SPECULAR reflection.

Specular is defined by the dictionary to mean “of, relating to, or having the properties of a mirror.”, so what we normally think of as reflection (like from a mirror) is in fact specular reflection.

There is non specular reflection, aka diffuse reflection, which just means that light is reflected off of a surface in a non mirror like way. This is accomplished with diffuse lighting where commonly we dot product the normal of a surface with the direction from the surface to the light to know how much to light that point on the surface.

If specular reflection is how mirrors reflect, and diffuse reflection is how regular diffuse surfaces work, then you might wonder what specular lighting is all about.

A specular highlight is actually just a cheap approximation to doing a mirror like specular reflection of a light source, so it is a cheap kind of specular reflection.

Let’s talk about how to do real mirror like specular reflection in a ray tracer.

When light hits a surface, some amount of the light will be reflected, and some amount of the light will be transmitted into the object. Transmitted light is used for the diffuse shading.

As you might imagine, the amount of light reflected plus the amount of light transmitted must equal the total amount of light that hit the surface. (note that some transmitted energy may be absorbed and given off as heat, or the object may be glowing, so may give off more light than received but let’s ignore that stuff for now.)

A common way to deal with this is to define a reflectivity of a surface as the amount of light it reflects in percent and use 100% minus that amount as the transmitted amount.

You might say that 2% of light that hits a surface reflects. That means that 98% of the light that hits a surface is transmitted and used for the diffuse shading.

When shading a point on the surface, you would calculate both the reflected color at that point on the surface, and the diffuse shaded color at that point, but you multiply the reflected color by 0.02 and the diffuse shaded color by 0.98 and add them together. Doing this gives a result like the below:

The higher the reflectivity percent, the more reflection you get, and the less diffuse shading you get. Turning down reflectivity has the opposite effect.

How do you calculate the reflected color of a point on a surface though?

You simply calculate the ray that reflected off of the surface, and calculate the color of what that ray hit.

To calculate a reflected ray, you need to know the direction that the ray was traveling when it hit the surface (The incident direction), you need to know the location that the ray hit the surface (the surface location), and you need to know the normal of the surface at the intersection point.

If you have those things you can calculate the reflected ray direction as this:

$ReflectRayLocation = SurfaceLocation \\ ReflectRayDirection = IncidentDirection - 2 * SurfaceNormal * DotProduct(IncidentDirection, SurfaceNormal)$

Note that in hlsl and glsl there is a function called “reflect” which takes the incident direction, and the surface normal, and returns you the reflected direction.

Doing the above is mathematically correct, but if you then try to raytrace from that ray, you may hit the same object you are reflecting off. One way to fight that problem is to push the ray positin a small amount away from the surface to make sure it misses it. You can do that for example like this:

$ReflectRayLocation = ReflectRayLocation + ReflectRayDirection * 0.01$

There are other ways to make sure you don’t hit the same object again, but pushing the ray away from the object a small amount really does work pretty nicely in practice.

# Refraction

I mentioned in the last section that whatever light wasn’t reflected when it hit a surface was transmitted. I also said that the transmitted light was used for diffuse shading.

In reality, it’s passed through the “bidirectional scattering distribution function” aka BSDF. You may have heard the term “bidirectional reflection distribution function” aka BRDF. A BRDF only deals with the hemisphere (half a sphere) that surrounds the surface normal. The BSDF on the other hand deals with the full sphere surrounding a surface normal so BRDF is a subset of what is possible with the BSDF.

Because the BSDF deals with an entire sphere, that means that it can handle reflection (specular and non specular) like the BRDF can, but it can also deal with transparency and refraction, where some of the light travels THROUGH an object.

In a path tracer where everything is physically accurate and mathematically precise, we would be interested in dealing with a BSDF and integrating over the sphere, but since we are working with a ray tracer, our physical accuracy needs are a lot lower – we only want a result that looks plausible.

What we are going to do for our transparency is calculate a direction that the ray is going to travel through the object, ray trace that ray to get a color, and use the transmitted light (the portion of light that isn’t reflected) as a multiplier of that color, that we add to the reflected amount of light.

If we have an object with 10% reflectivity, and 90% transmittance, but use that transmitted light for transparency, we’ll have a rendering like below:

Now that we have transparency, let’s talk about refraction. Refraction is a physical phenomenon where light bends (“changes speed” i guess but that sounds a bit suspicious for light), as it hits a boundary between two different surfaces.

Every material has a refractive index, and in fact, may have different refractive indices per light frequency. For our purposes, we’ll assume surfaces have the same refractive index for every frequency of light. There’s a list of refractive indices for a lot of different materials here: Index of refraction

How a ray bends when it refracts depends on the ration of the refractive index that it’s leaving to the refractive index that it’s entering. AKA outside/inside.

HLSL and GLSL have a function called refract which take the incident vector, the surface normal vector, and that ration of refractive indices, and return the refracted ray.

When you do a raytrace down the refracted ray, you will have the same problem as when tracing the reflected ray, that you may hit the same surface you just hit again erroneously. To help that, you once again just move the ray slightly away from the surface.

$RefractRayLocation = RefractRayLocation + RefractRayDirection * 0.01$

Here is a rendering where the sphere has 10% reflectivity, 90% transmittance, an air refractive index of 1.0, and a refractive index of 1.125 for the sphere. You can see how the light bends as it goes through the object and looks pretty neat!

# Fresnel

There is an interesting property in our world: If you look at something straight on, it’s the least reflective it will be. If you turn it nearly 90 degrees where it’s nearly edge on, it will be the most reflective it can be. Many surfaces will become almost perfectly reflective when you view them almost edge on. Go try it out with a wall in your house or a glass or other things.

Weird huh? That’s called Fresnel, and is something we can also make use of in ray tracing to get a more realistic image.

Instead of just always using the reflectivity amount of the surface, we use the Fresnel equation to figure out how much reflectivity an object should have based on the view angle versus the surface normal, so that when it’s more edge on it becomes more reflective. At minimum the reflectivity will be the reflectivity of the surface, and at maximum the reflectivity will be 100%.

The amount we transmit for either refraction or diffuse will be 100% minus however much percentage is reflective.

Here is the image showing normal reflection again:

Here is the image with Fresnel:

It looks quite a bit better with fresnel doesn’t it?!

Here’s a GLSL function of Schlick’s Fresnel approximation function. Notice that it takes the surface normal and incident vector, as well as the refractive index being left (n1) and the refractive index being entered (n2):

```float FresnelReflectAmount (float n1, float n2, vec3 normal, vec3 incident)
{
// Schlick aproximation
float r0 = (n1-n2) / (n1+n2);
r0 *= r0;
float cosX = -dot(normal, incident);
if (n1 > n2)
{
float n = n1/n2;
float sinT2 = n*n*(1.0-cosX*cosX);
// Total internal reflection
if (sinT2 > 1.0)
return 1.0;
cosX = sqrt(1.0-sinT2);
}
float x = 1.0-cosX;
float ret = r0+(1.0-r0)*x*x*x*x*x;

// adjust reflect multiplier for object reflectivity
ret = (OBJECT_REFLECTIVITY + (1.0-OBJECT_REFLECTIVITY) * ret);
return ret;
}
```

Our tale of reflection is complete, so let’s go back to refraction / transparency and finish up the last two items.

# Total Internal Reflection

The way that the Fresnel equation works, it’s possible that when moving from a material with a higher index of refraction to a lower one, that the amount of refraction can actually drop to zero percent. In this case, the light doesn’t exit the higher refractive index object and instead ONLY reflects back into the object, because the reflective amount becomes 100%.

When this happens, it’s called “Total Internal Reflection” for hopefully obvious reasons (:

There isn’t a whole lot to say about this, because you can see that this is even accounted for in the GLSL Fresnel function from the last section.

However, if you are ever under water in a swimming pool and look up to see the water surface looking like a mirror, that is total internal reflection in action.

You can also see it in the render below, where you can see reflections on the inside of the walls of the object, especially on the bottom (floor) of the object:

# Beer’s Law

Beer’s law is the last item to talk about, and relates to transparent surfaces. Beer’s law deals with light being absorbed over distance as it travels through a material.

Beer’s law is why a thin piece of jello is mostly colorless, but a thicker piece of jello has a much richer and deeper color.

Here’s a cube with beer’s law absorbing red and green light over distance. You should notice that where the light travels less distance through the cube that it’s not as blue, because not as much red and green light has been absorbed:

To apply beer’s law, you first figure out how long a ray has traveled through the absorbing material (by tracing the ray inside the object to see where it hits the back side). You then do this calculation:

```vec3 color = RayTrace(rayPos, rayDir);
vec3 absorb = exp(-OBJECT_ABSORB * absorbDistance);
color *= absorb;
```

OBJECT_ABSORB is an RGB value that describes how much of each color channel absorbs over distance. For example, a value of (8.0, 2.0, 0.1) would make red get absorbed the fastest, then green, then blue, so would result in a blueish, slightly green color object.

# Putting it All Together

Now that we have the individual components worked out let’s talk about how to put it together.

Firstly, when a ray hits any surface, you need to use the Fresnel equation to get the multiplier for the reflected and transmitted light.

Next you calculate the reflected and transmitted light by recursively raytracing. The transmitted light is either used for the diffuse shading, the transparency/refracted ray, or some combination of those two (technically, it’s all up to the BSDF we are approximating).

Then, you multiply the reflected light by the reflected amount from the Fresnel equation, and the transmitted amount by 100%-reflectionAmount and add the results together.

(Quick side note, if you have emissive color on the surface aka the object glows, you would also add that in here).

Since raytracing is recursive, you would do this each time a ray intersected with an object. In other words, each intersection causes the ray to split into two rays.

As you can imagine, this can make the rendering quite complex, especially on the GPU where you can’t even do real recursion.

One way to help limit the recursiveness a bit is when you are calculating your reflection and transmittance amounts, you can choose a threshold like say 1% where if the reflection is under 1% it clamps it to 0%, and if it’s greater than 99% it clamps it at 100%. You can then choose not to recurse down a specific ray if the ray’s multiplier is 0. The end result will be that reflections or transmittance rays that don’t contribute much to the end result won’t be followed at all.

If you are willing to sacrifice some visual quality to not have to split your ray into two at each object intersection, you could also figure out if reflection or transmittance has the higher multiplier, and make the ray only follow one of the paths. If you were doing this in a path tracer, you could choose which one to follow randomly, using the multiplier as a weight for the random selection.

The problem in both of these two optimizations is that the multiplier is only half of the information though so may incorrectly choose the less meaningful contribution. The other half of the information is the color of the ray if you followed it. The reason for this is that you might have a low multiplier for a really bright spot (caustics have this problem commonly!), or vice versa you may have a high multiplier for a dull featureless spot. With path tracing, if you take enough samples, it all washes out in the averages, and with ray tracing, maybe you accept that it will do the wrong thing sometimes, to stay a real time algorithm, but it’s important to know how this type of choice can fail for you. (Side note, this sort of stuff is what importance sampling in path tracing is all about – trying to make rays follow more meaningful paths to get better results quicker).

When doing real time raytracing you also often have to decide how many times you want to allow a ray to bounce around. In the shadertoy that goes with this post, that parameter is MAX_RAY_BOUNCES and I have it set to 10.

Interestingly, setting it to 1 has no visible impact on the sphere at all, which is a nice improvement. For the box, a value of 3 seems to be the maximum it needs. 3 also seems to be the magic number for the geometric gem type shape.

So, 10 is overkill, but i left it at that in case people play with parameters and change them to values which would require more bounces.

Lastly I wanted to mention that in the scene that I rendered, I did a small “trick” to make it so I didn’t need to do full recursive splitting of rays at each intersection. I did this by making it so the main object in the center of the scene was the only object that had reflection.

This way, I only need to split the ray into two if i hit the main object. Furthermore, when I’m splitting the ray at the main object, the ray that gets the color for the outside world (versus the inside of the object) is a single non recursive ray cast since it can’t hit anything reflective. The result is that at each intersection of the sphere, i do a simple non recursive ray cast for the ray that is going outside of the main object, then i continue the iterative ray on the inside of the object until i run out of bounces.

Doing this causes a recursive process to become an iterative one, which is much friendlier on the gpu.

Below is the final render again from the shadertoy. The parameters are:

• The refractive index of the air is 1.00029
• The refractive index inside the objects are 1.125
• Reflectivity is 1%
• The absorption for beers law is (8.0, 8.0, 3.0)