1. A consequence of Fermat’s principle from optics is that light traveling from a point normal p 1 in a medium with index of refraction eta 1 to a point normal p 2 in a medium with index of refraction eta 2 will follow a path that minimizes the time to get from the first point to the second point. Snell’s law can be shown to follow from this fact directly. Consider light traveling between two points normal p 1 and normal p 2 separated by a planar boundary. The light could potentially pass through the boundary while traveling from normal p 1 to normal p 2 at any point on the boundary (see Figure 8.25, which shows two such possible points normal p prime and normal p double-prime ). Recall that the time it takes light to travel between two points in a medium with a constant index of refraction is proportional to the distance between them times the index of refraction in the medium. Using this fact, show that the point normal p prime on the boundary that minimizes the total time to travel from normal p 1 to normal p 2 is the point where eta 1 sine theta 1 equals eta 2 sine theta 2 .
    Figure 8.25: Derivation of Snell’s Law. Snell’s law can be derived using Fermat’s principle, which says that light will follow the path that takes the least amount of time to pass between two points. The angle of refraction theta at the boundary between two media can thus be shown to be the one that minimizes the time spent going from normal p 1 to a point normal p Subscript on the boundary plus the time spent traveling the distance from that point to normal p 2 .
  2. Read the papers of Wolff and Kurlander (1990) and Tannenbaum, Tannenbaum, and Wozny (1994), and apply some of the techniques described to modify pbrt to model the effect of light polarization. Set up scenes and render images of them that demonstrate a significant difference when polarization is accurately modeled.
  3. Construct a scene with an actual geometric model of a rough plane with a large number of mirrored microfacets, and illuminate it with an area light source. Place the camera in the scene such that a very large number of microfacets are in each pixel’s area, and render images of this scene using hundreds or thousands of pixel samples. Compare the result to using a flat surface with a microfacet-based BRDF model. How well can you get the two approaches to match if you try to tune the microfacet BRDF parameters? Can you construct examples where images rendered with the true microfacets are actually visibly more realistic due to better modeling the effects of masking, self-shadowing, and interreflection between microfacets?
  4. Extend pbrt to be able to more accurately render interesting surfaces like wood (Marschner et al. 2005), cloth (Sattler et al. 2003), or car paint (Günther et al. 2005). Render images showing better visual results than when existing reflection functions in pbrt are used for these effects.
  5. Implement a simulation-based approach to modeling reflection from complex microsurfaces, such as the one described by Westin, Arvo, and Torrance (1992). Modify pbrt so that you can provide a description of the microgeometry of a complex surface (like cloth, velvet, etc.), fire rays at the geometry from a variety of incident directions, and record the distribution and throughput for the rays that leave the surface. (You will likely need to modify the PathIntegrator from Chapter 14 to determine the distribution of outgoing light.) Record the distribution in a 3D table if the surface is isotropic or a 4D table if it is anisotropic, and use the table to compute BRDF values for rendering images. Demonstrate interesting reflection effects from complex surfaces using this approach. Investigate how the size of the table and the number of samples taken to compute entries in the table affect the accuracy of the final result.
  6. Although pbrt features a Curve shape that provides fairly efficient intersection tests between rays and parametric curves (Section 3.7), it lacks a reflection model for hair. Choose one of the models described in the “Further Reading” section such as Marschner et al.’s (2003) or d’Eon et al.’s (2011), and implement it in pbrt. Either find a geometric model of hair or generate hair procedurally, and render images using your implementation.