Fundamentals of Computer Graphics, 3rd Edition

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13

More Ray Tracing

A ray tracer is a great substrate on which to build all kinds of advanced rendering

effects. Many effects that take significant work to fit into the object-order ras-

terization framework, including basics like the shadows and reflections already

presented in Chapter 4, are simple and elegant in a ray tracer. In this chapter we

discuss some fancier techniques that can be used to ray-trace a wider variety of

scenes and to include a wider variety of effects. Some extensions allow more gen-

eral geometry: instancing and constructive solid geometry (CSG) are two ways

to make models more complex with minimal complexity added to the program.

Other extensions add to the range of materials we can handle: refraction through

transparent materials, like glass and water, and glossy reflections on a variety of

surfaces are essential for realism in many scenes.

This chapter also discusses the general framework of distribution ray trac-

ing (Cook et al., 1984), a powerful extension to the basic ray-tracing idea in which

multiple random rays are sent through each pixel in an image to produce images

with smooth edges and to simply and elegantly (if slowly) produce a wide range

of effects from soft shadows to camera depth-of-field.

If you start with a brute-

force ray intersection loop,

you’ll have ample time to

implement an acceleration

structure while you wait for

images to render.

The price of the elegance of ray tracing is exacted in terms of computer time:

most of these extensions will trace a very large number of rays for any non-trivial

scene. Because of this, it’s crucial to use the methods described in Chapter 12 to

accelerate the tracing of rays.

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304 13. More Ray Tracing

13.1 Transparency and Refraction

In Chapter 4 we discussed the use of recursive ray tracing to compute specular,

or mirror, reflection from surfaces. Another type of specular object is a dielec-

tric—a transparent material that refracts light. Diamonds, glass, water, and air are

dielectrics. Dielectrics also filter light; some glass filters out more red and blue

light than green light, so the glass takes on a green tint. When a ray travels from

a medium with refractive index n into one with a refractive index n

t

, some of the

light is transmitted, and it bends. This is shown for n

t

>nin Figure 13.1. Snell’s

law tells us that

n sin θ = n

t

sin φ.

Computing the sine of an angle between two vectors is usually not as convenient

Example values of

n

:

air: 1.00;

water: 1.33–1.34;

window glass: 1.51;

optical glass: 1.49–1.92;

diamond: 2.42.

as computing the cosine, which is a simple dot product for the unit vectors such

as we have here. Using the trigonometric identity sin

2

θ +cos

2

θ =1, we can

derive a refraction relationship for cosines:

cos

2

φ =1−

n

2



1 − cos

2

θ



n

2

t

.

Note that if n and n

t

are reversed, then so are θ and φ as shown on the right of

Figure 13.1.

To convert sin φ and cos φ into a 3D vector, we can set up a 2D orthonormal

basis in the plane of the surface normal, n, and the ray direction, d.

From Figure 13.2, we can see that n and b form an orthonormal basis for the

plane of refraction. By definition, we can describe the direction of the transformed

Figure 13.1. Snell’s Law describes how the angle φ depends on the angle θ and the

refractive indices of the object and the surrounding medium.