News (August, 31): We are working on Scratchapixel 3.0 at the moment (current version of 2). The idea is to make the project open source by storing the content of the website on GitHub as Markdown files. In practice, that means you and the rest of the community will be able to edit the content of the pages if you want to contribute (typos and bug fixes, rewording sentences). You will also be able to contribute by translating pages to different languages if you want to. Then when we publish the site we will translate the Markdown files to HTML. That means new design as well.
That's what we are busy with right now and why there won't be a lot of updates in the weeks to come. More news about SaP 3.0 soon.
We are looking for native Engxish (yes we know there's a typo here) speakers that will be willing to readproof a few lessons. If you are interested please get in touch on Discord, in the #scratchapixel30 channel. Also looking for at least one experienced full dev stack dev that would be willing to give us a hand with the next design.
Feel free to send us your requests, suggestions, etc. (on Discord) to help us improve the website.
And you can also donate). Donations go directly back into the development of the project. The more donation we get the more content you will get and the quicker we will be able to deliver it to you.
The other advantage of raytracing is that, by extending the idea of ray propagation, we can very easily simulate effects like reflection and refraction, both of which are handy in simulating glass materials or mirror surfaces. In a 1979 paper entitled "An Improved Illumination Model for Shaded Display", Turner Whitted was the first to describe how to extend Appel's raytracing algorithm for more advanced rendering. Whitted's idea extended Appel's model of shooting rays to incorporate computations for both reflection and refraction.
In optics, reflection and refraction are wellknown phenomena. Although a whole later lesson is dedicated to reflection and refraction, we will look quickly at what is needed to simulate them. We will take the example of a glass ball, an object which has both refractive and reflective properties. As long as we know the direction of the ray intersecting the ball, it is easy to compute what happens to it. Both reflection and refraction directions are based on the normal at the point of intersection and the direction of the incoming ray (the primary ray). To compute the refraction direction we also need to specify the index of refraction of the material. Although we said earlier that rays travel in a straight line, we can visualize refraction as the ray being bent. When a photon hits an object of a different medium (and thus a different index of refraction), its direction changes. The science of this will be discussed in more depth later. As long as we remember that these two effects depend on the normal vector and the incoming ray direction and that refraction depends on the refractive index of the material we are ready to move on.
Similarly, we must also be aware of the fact that an object like a glass ball is reflective and refractive at the same time. We need to compute both for a given point on the surface, but how do we mix them? Do we take 50% of the reflection result and mix it with 50% of the refraction result? Unfortunately, it is more complicated than that. The mixing of values is dependent upon the angle between the primary ray (or viewing direction) and both the normal of the object and the index of refraction. Fortunately for us, however, there is an equation that calculates precisely how each should be mixed. This equation is know as the Fresnel equation. To remain concise, all we need to know, for now, is that it exists and it will be useful in the future in determining the mixing values.
So let's recap. How does the Whitted algorithm work? We shoot a primary ray from the eye and the closest intersection (if any) with objects in the scene. If the ray hits an object which is not a diffuse or opaque object, we must do extra computational work. To compute the resulting color at that point on, say, for example, the glass ball, you need to compute the reflection color and the refraction color and mix them. Remember, we do that in three steps. Compute the reflection color, compute the refraction color, and then apply the Fresnel equation.

First we compute the reflection direction. For that, we need two items: the normal at the point of intersection and the primary ray's direction. Once we obtain the reflection direction, we shoot a new ray in that direction. Going back to our old example, let's say the reflection ray hits the red sphere. Using Appel's algorithm, we find out how much light reaches that point on the red sphere by shooting a shadow ray to the light. That obtains a color (black if it is shadowed) which is then multiplied by the light intensity and returned to the glass ball's surface.

Now we do the same for the refraction. Note that, because the ray goes through the glass ball it is said to be a transmission ray (light has traveled from one side of the sphere to the other; it was transmitted). To compute the transmission direction we need the normal at the hit point, the primary ray direction, and the refractive index of the material (in this example it may be something like 1.5 for glass material). With the new direction computed, the refractive ray continues on its course to the other side of the glass ball. There again, because it changes medium, the ray is refracted one more time. As you can see in the adjacent image, the direction of the ray changes when the ray enters and leaves the glass object. Refraction takes place every time there's a change of medium and the two media, the one the ray exits from and the one it gets in, have a different index of refraction. As you probably know the refraction index of air is very close to 1 and the refraction index of glass is around 1.5). Refraction has for effect to bend the ray slightly. This process is what makes objects appear shifted when looking through or at objects of different refraction indexes. Let's imagine now that when the refracted ray leaves the glass ball it hits a green sphere. There again we compute the local illumination at the point of intersection between the green sphere and refracted ray (by shooting a shadow ray). The color (black if it is shadowed) is then multiplied by the light intensity and returned to the glass ball's surface

Lastly, we compute the Fresnel equation. We need the refractive index of the glass ball, the angle between the primary ray, and the normal at the hit point. Using a dot product (we will explain that later), the Fresnel equation returns the two mixing values.
Here is some pseudocode to reinforce how it works:
One last, beautiful thing about this algorithm is that it is recursive (that is also a curse in a way, too!). In the case we have studied so far, the reflection ray hits a red, opaque sphere and the refraction ray hits a green, opaque, and diffuse sphere. However, we are going to imagine that the red and green spheres are glass balls as well. To find the color returned by the reflection and the refraction rays, we would have to follow the same process with the red and green spheres that we used with the original glass ball. This is a serious drawback of the ray tracing algorithm and can be nightmarish in some cases. Imagine that our camera is in a box that has only reflective faces. Theoretically, the rays are trapped and will continue bouncing off of the box's walls endlessly (or until you stop the simulation). For this reason, we have to set an arbitrary limit that prevents the rays from interacting, and thus recursing endlessly. Each time a ray is either reflected or refracted its depth is incremented. We simply stop the recursion process when the ray depth is greater than the maximum recursion depth.