It is so interesting breaking down each type of reflection. It really allows me to take a step back and appreciate rendered images and video games for the amount of detail that goes into every object. Looking at some of the games that Rockstar makes, I can't understand how complicated the system is and how they built such a world that reflects the realities of the real world.

I'm wondering if there are any simple mathematical models for these different categories of reflection functions that are easy to compute in real time.

The main reason for the moon's retro-reflectivity is a layer of dust on the moon's surface, called the regolith. The Apollo 11 mission also put retro-reflectors on the moon as well.

Is there a reason why certain materials give different reflections compared to other materials?

If you've wondered how retro-reflective material works, one way to implement it is to cover the surface of the material with repeating microscopic corners. This is commonly used in street signs so that headlights can illuminate them from any angle back onto the driver.

Here is an interesting found of reflection. Where the wet floor is considered a reflection. https://handwiki.org/wiki/Wet_floor_effect

I wonder what exactly determines what category of reflections a given object belongs to? In other words, which reflection function should we use for an object if we are just given its model and material? Intuitively I think its surface smoothness probably plays a big role in this, but it there anything else? And also how do we measure an object's "reflectivity" and reflect (hehe) that in our code and renders?

I believe the material is the key point that determines the category of reflections. For example, the perfect mirror reflection is a ideal speular. However, there is no perfect mirror in the world, so most reflection is the mixture of many categories.

@Sicheng-Pan I was wondering the same thing, because looking back in retrospect at some of the more hyper-realistic video games I've played, I can only imagine who much complexity is involved in determining the amount of reflectivity of each individual object. Intuitively, I feel like there has to be a quick way to compute some of them, otherwise the games would not be able to run smoothly or in high resolution on consoles that have limited RAM.

@waleedlatif Looking back on the project, I think the "quick" way to compute the reflectivity is to pre-load the value in as a property or a function of angles/vectors, and then retrieve those values when necessary to compute the actual amount of light reflected, rather than having to the compute the reflectance itself each time