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 directly from this fact.
    Figure 9.56: 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 .
    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 9.56, 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 .
  2. Read the recent paper by d’Eon (2021) that describes a BRDF based on a model of the aggregate scattering of large collections of spherical particles that are themselves Lambertian. Implement this approach as a new BxDF in pbrt and render images comparing its visual appearance to that of the DiffuseBxDF.
  3. Read the paper of Wolff and Kurlander (1990) and the course notes of Wilkie and Weidlich (2012) and apply some of the techniques described to modify pbrt to model the effect of light polarization. (A more in-depth review of the principles of polarization is provided by Collett (1993).) Set up scenes and render images of them that demonstrate a significant difference when polarization is accurately modeled. For this, you will need to implement a polarized version of the Fresnel equations and add BSDFs that model optical elements like linear polarizers and retarders.
  4. 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?
  5. One shortcoming of the microfacet-based BSDFs in this chapter is that they do not account for multiple scattering among microfacets. Investigate previous work in this area, including the stochastic multiple scattering model of Heitz et al. (2016b) and the analytic models of Lee et al. (2018) and Xie and Hanrahan (2018), and implement one of these approaches in pbrt. Then implement an approximate model for multiple scattering, such as the one presented by Kulla and Conty Estevez (2017) or by Turquin (2019). How do rendered images differ from pbrt’s current implementation? How closely do the approximate approaches match the more comprehensive ones? How does execution time compare?
  6. Review the algorithms for efficiently finding an approximation of a material’s normal distribution function and using that to measure BRDFs that are outlined in Section 9.8 and explained in more detail in Dupuy and Jakob (2018). Follow this approach to implement a virtual gonioreflectometer, where you provide pbrt with a description of the microgeometry of a complex surface (cloth, velvet, etc.) and its low-level reflection properties and then perform virtual measurements of the BSDF by simulating light paths in the microgeometry. Store the results of this simulation in the file format used by the MeasuredBxDFData and then render images that compare using the tabularized representation to directly rendering the microgeometry. How do the images compare? How much more computationally efficient is using the MeasuredBxDFData?
  7. Marschner et al. (2003) note that human hair actually has an elliptical cross section that causes glints in human hair due to caustics; subsequent work by Khungurn and Marschner (2017) proposes a model that accounts for this effect and shows that it matches measurements of scattering from human hair well. Extend the HairBxDF implementation here, following their approach. One issue that you will need to address is that the partial-differential normal p slash partial-differential v returned by Curve::Intersect() is always perpendicular to the incident ray, which leads to different orientations of the azimuthal coordinate system. This is not an issue for the model we have implemented, since it operates only on the difference between angles phi in the hair coordinate system. For elliptical hairs, a consistent azimuthal coordinate system is necessary.
  8. Read Yan et al.’s paper on fur scattering (2015) and implement their model, which accounts for scattering in the medulla in fur. Render images that show the difference from accounting for this in comparison to the current implementation. You may want to also see Section 4.3 of Chiang et al. (2016a), which discusses extensions for modeling the undercoat (which is shorter and curlier hair underneath the top level) and a more ad hoc approach to account for the influence of scattering from the medulla.
  9. Read one or more papers from the “Further Reading” section of this chapter on efficiently rendering glints, which are evident when the surface microstructure is large enough or viewed closely enough that the assumption of a continuous distribution of microfacet orientations is no longer valid. Then, choose one such approach and implement it in pbrt. Render images that show off the effects it is capable of producing.