We will start off with the Law of Reflection, which states that the angle of incidence is equal to the angle of reflection. But what does this mean geometrically? Let’s start with a flat surface. A line that is perpendicular to the surface is called a "normal" to the surface, and the incoming light creates an angle relative to that line. That angle is the angle of incidence. The Law of Reflection tells us that the reflected light will form an angle with the normal that is equal to the angle of incidence.

All reflected light will follow the law of reflection, but what we see will greatly depend on the nature of the surface that is doing the reflecting.

The simplest case to consider is a flat mirror. In this case, the normal to the surface is always pointing in the same direction, which creates a simple geometry that is easy to understand.

If you are looking at yourself in a mirror, the reflection you see is formed by light bouncing off of you, hitting the mirror, and then hitting your eyes. But your eyes don’t know the entire path that the photons have taken. They can only see the last step of the journey, which is the light bouncing off the mirror and reaching your eyes. Your brain takes the incoming photons and imagines that those photons were coming straight towards you the whole time. What your brain sees is the "image" of your body.

With a flat mirror, that image is literally a reflection of yourself as if you were on the other side of the mirror (and flipped). This is why flat mirrors behave like windows into some alternate universe where everything is flipped backwards. This is all due to the geometry of the flat mirror.

The reason that curved mirrors create a distorted picture of reality is that the geometry of the mirror creates an warped alternate reality on the other side of it.

A convex mirror is one that is curved outward. A typical convex mirror is the round mirror that you see in stores and restaurants that allow someone to see around corners easily. When you look into it, you see a version of reality that feels a little bit squished. We can make sense of this by looking at the geometry.

For convex mirrors, the normal to the surface changes based on the location. The part that is closest to you will have a normal that is pointing right back at you. But as you move away from that point, the normal points outward. So as you shift your eyes upward along the surface, the normal vector is also pointing upward. If we apply the Law of Reflection to that point, we’ll see that it exaggerates the shift even more. So if you look slightly upward from the closest point, you get a reflection that is from much further up than what you would see with a flat mirror.

The image that your brain constructs is therefore a compressed version of reality. If you would have to look up a certain amount to see the top of your head with a flat mirror, you would have to look up less with a convex mirror, and so the image of your head is going to be compressed.

Concave mirrors are mirrors that are bent inward. Concave mirrors can give two different types of images, depending on how close or far you are relative to the curvature of the mirror. Let’s start with a very slightly curved mirror. As the mirror gets curved inward, your image will start to grow taller. This is because the normal to the surface is pointing in the opposite direction that your eyes are moving (relative to the nearest point). As you look upward, the normal points downward, which means that the movement is lessened. You would have to look up further to see the top of your head than you would with a flat mirror.

However, if the mirror is curved enough, you will actually start to see an upside down version of yourself. The reason is that the normal to the surface changes so much that you have to look downward to get a reflection from the upwards direction.

Funhouse mirrors are a combination of convex and concave, which leads to the alternating behavior of stretching and squishing.

Rough surfaces reflect light in essentially random ways. This means that the light reaching our eyes from a certain place are coming from all sorts of places. This leads to the image being created from random places in the room, which leads to the absence of any type of discernable image.