A Gentle Introduction to the Rendering Equation

August 11, 2022

I’ve been playing around with Blender lately, and it’s amazing how software can produce incredibly realistic images from 3D scenes.

Notice the effects in the image below.

The ripples in the water, smoke coming out of the chimney, the glow from the fireplace in the cabin. All of these are computer-generated.

Blender has a powerful built-in renderer called Cycles that produced this image. The artist created the 3D models, specified their materials, added some details and lighting, and Cycles generated the 2D image.

What if you wanted to write your own renderer?

You'd need to first model the way that light moves in 3D space. You'd need a rendering equation.

The equation above is the Rendering Equation, which calculates the light leaving a point on an object in a certain direction.

We'll first get a feel for how this works, and then go over it in detail.

Why is the wall orange?

Let's take a deeper look at one spot on the image above: the glow from the fireplace on the interior wall.

Here we have a 3D scene that includes just the two walls of the cabin and the fireplace.

Imagine a beam shooting out from the bright fire in the fireplace, bouncing off the wall, and hitting our eye.

The yellow light of the fire travels to the wall, mixes with the brown wood, and a portion of it bounces back toward our eye to give us an orange colour.

The Rendering Equation tells us exactly how the properties of the wall, the fire, and the angles of the bounces combine to give us this colour.

Building some intuition

Let’s go over some concepts that can help you intuitively grasp the Rendering Equation.

How light can hit your eye

There are three ways that light can hit your eye:

1. Light directly hits your eye: the light from a lightbulb 💡
2. Light refracts: the sun shining through a glass window 🪟
3. Light reflects: the sun reflecting off of a car 🚙

Everything reflects

When we think of reflecting, we often think mirrors, but every object reflects light to some degree. If you’re reading a book on a sunny day, you’re reflecting a little bit of light onto your face.

Material matters

Imagine you're holding a mirror in your hands angled towards your face on a sunny day. This is going to shine a lot more light onto your face than if you were reading a book.

Materials affect the light that's reflected off a surface.

The equation

Okay, now onto the equation.

This calculates the outgoing light from a point on a surface.

Put more simply: the light that hits your eye = the light emitted from the object + light hitting the surface from every 3D direction weighted by some calculations having to do with the material and the angle.

Light hitting your eye

This is what we're calculating, the light from a point on a surface in the direction toward your eye.

Light emitted from the object

An object itself can emit light, like a fire, or a piece of metal that gets really hot.

Incoming light

When we think of light bouncing, we might think it comes in at the same direction that it leaves, like a mirror. But for any realistic surface, there are many light beams that will exit in a different angle.

This means the light reflected from a surface into our eye could have hit the surface at any angle. So, to account for all this light, we have to consider every incoming direction.

The material function

The material function considers many properties of the material and how they affect how the incoming light reflects into outgoing light.

Materials (like metal and paper) have different properties that affect how light is reflected and refracted. A mirror might reflect a lot of light, with the same colour as the incoming light.

Properties include roughness, transmission (high for glass), and specularity (low for matte surfaces).

The dot product

The difference between the surface normal and the angle of the incoming light affects how much light is reflected.

If the incoming light angle is close to the normal, the amount of light will be higher.

Back to the orange spot

Now that we have an understanding of how the Rendering Equation works, we can look at the interior wall example again.

Light hitting your eye

We're calculating the light from the spot on the wall to our eye.

Light emitted from the object

The wall itself doesn't emit any light. The fire emits light, which increases the amount of incoming light onto the wall.

Incoming light

We have the strong warm-coloured incoming light from the fire, as well as light bouncing off the surrounding walls to hit this one spot.

The material function

The brown wood stain on the wall absorbs some light wavelengths and reflects other wavelengths, which gives the reflected light a more brown tinge. The wood is rough, so it doesn't always reflect light at a mirrored angle.

The dot product

The wall is directly facing the fire, so it receives a lot of its light.

The result

And our final colour is a fairly bright orange from the strong yellow light mixing with the brown stain on the wooden wall.

Questions

• How is colour represented in this equation?

  We've been talking about how the rendering equation measures "light." More specifically, what it measures is something called spectral radiance. Put simply, spectral radiance is a representation of light that encodes data that gives us the colour and intensity.

• What about refraction?

  The equation above just focuses on reflection, but can be expanded to include refraction. Instead of integrating over the unit hemisphere, we'd integrate over the unit sphere, and the material function would account for transmission of light.

• What other techniques do renderers implement to solve the rendering equation?

  Renderers may sample a pixel many times to produce the most accurate colour, ignore objects that are far away, and use integrated hardware like GPUs to speed things up.

Further reading

• Read about how Cycles works
• Build your own C++ ray tracer with "Ray Tracing in one Weekend"
• A more in-depth overview of ray tracing