“Bidirectional Reflectance Distribution Function”

In the past, there were no such fancy words or concepts in our lives. The only thing that we did, were playing with the options of the reflection channel. Whenever the physical render engines included our lives, then we changed our point of view to the world; and now we started looking at a metallic surface or even a table surface with very careful eyes. Because of these realistic render engines, we decided that we need to know the laws behind the appearance. We managed well with these realistic engines for a while. But when we decided to swim in the water we did not know, and when we dived deep into that water, the mathematical complexity of the physical world appeared. What we previously called Reflection suddenly became a "Reflectance". For years, the reflection we know as a friend turned into a "mathematical monster." We have to face such these mathematical monsters. Our adventure has just begun.

Now let's try to scare you a little bit before we get into the matter: for example, look at the figure below:

or look at this:

If you were not afraid enough, we kept the worst at the end: Now look at the formula below:

It's all about BRDF and reflection that will come out if you go deep enough. It's one of the "Mathematical Monster" that we mentioned. Now, beside the fearless mathematicians who are not afraid to see this, let's try to make things a little easier for 3D artists with mortal eyes.

Well let's ask again: What is BRDF?

BRDF is actually a formulated function that you often see in reflection models we have used for years. Like the above formula. So, in fact BRDF is a function and this function is defined as the ratio of the reflected light (radiance energy) to the incoming light (irradiance energy) at the point "x" on the surface.

BRDF is one of the most useful models of reflection and was first described in the field of Radiometry. It deals with the directional distribution reflection of light from any surface. It does not deal with the part of the light that travels into the surface, it only deals with the reflected part. In fact, BRDF is a simplified version of the "Bidirectional Scattering Surface Reflection Distribution Function (BSSRDF)". In BSSRDF, it is also taken into account which point the light enter into the surface and where it left off from the surface. In BRDF the assumption is that the surface is homogeneous. Thus, the BSSRDF is reduced to BRDF.

The fact that we have known for many years as "reflection" has already been described as a function. Only these reflection models have changed over the years. We used this function for years, but the name was "Phong" or "Lambertian" or "Blinn-Phong" before. These were relatively simple models that did not produce physical results. We are now using BRDF models, such as Torrence-Sparrow or Cook-Torrence, which are based on years ago and produce satisfactory results on physical renders.

A BRDF must obey with the basic laws of physics in order to give realistic results. Let's look at these briefly:

Energy conservation

Energy conservation must be ensured in any system. Therefore, a good and realistic BDRF must obey with the law of conservation of energy. Accordingly, for all possible directions, the total energy of the light reflected from a surface can never be more than the total energy of the incoming light. We will show this later using the new Octane BDRF models.


The reflection value must be between 0 and 1 because of the conservation of the energy. Therefore, the ratio of the reflected light (radiance energy) to the incoming light (irradiance energy) must be between 0 and 1. BRDF also includes this ratio with the cosine term. That is, the range of BRDF is 0 to infinity. Also both radiance and irradiance values can not be negative. Therefore BRDF can not take negative value either.


This feature is also known as Helmotz's reciprocity law. This law is caused by another physical characteristic of the light. According to this law; the BRDF value must not change when the incoming vector and the outgoing (view) vector are interchanged. In other words, when the travel of the light reflected from the surface is reversed, the light must follow the same path and the BRDF must be the same in both cases.


Microfacet theory is a technique developed on the basis of roughness or imperfection of every surface in the real world (let's call it Roughness). Works in cooperation with shadowing and masking. In realistic BRDF models, the surface is considered to be formed from uneven grooves (microfacet). Microfacet is calculated by how much of the ratio is due to masking and shadowing of interaction with the light.

The masking state is, when the light is reflected by one surface of the microfacet, then the reflected portion is blocked by the other "microfacet". Shadowing is the case where the light is blocked by the other "microfacet" before it reaches the surface of the "microfacet".


A good BRDF should also present the Fresnel Equation. Fresnel is the name given to the fact that the surface reflection varies according to the observed angle and the surface IOR (Index of Refraction) values. The Fresnel effect is, which reflects less light from the direct-facing surfaces than other angles (reflection changes when you look at different angles other then grazing angle). For example, if you look at the plant pot on a reflective table in orthogonal direction (steep angles), you will see a weak reflection. But if you move closer to the Shallow Angle (grazing angle), you will see stronger reflections. So Grazing Angle explains why most materials are actually "The Reflectors". The Fresnel effect takes an important place in creating realistic material.

isotropy and anisotropy BRDF

Isotropic and Anisotropic BRDFs are an important subclass of BRDF. For many materials in the nature, if the direction of the light and the view direction are fixed, the reflection does not change as the material is rotated around the surface normal. These properties are called isotropic materials. Examples of isotropic materials include metals, plastics and painted surfaces.

Anisotropic surfaces, on the other hand, change their reflectance values by rotating the surface around its normal, in contrast to isotropic surfaces. Examples of anisotropic surfaces are metal brushes, polished metal, human hair, fur, velvet and wood.


We tried to talk BRDF here simply, but this is a very complex and long topic, you can get more information by looking at the links below.

An Overview of BRDF Models - here

Wikipedia page about BRDF - here

About microfacet method - here

BRDF for computer Graphics - here

Wikipedia page for Fresnel Equations - here