Book 3 review, Chapters 4-6, responding to feedback

Author: trevordblack

books/RayTracingTheRestOfYourLife.html

@@ -1309,10 +1309,10 @@
 least for ray tracing--is producing a random direction. In the first two books we generated a random
 direction by creating a random vector and rejecting it if it fell outside of the unit sphere. We
 repeated this process until we found a random vector that fell inside the unit sphere. Normalizing
-this vector produced points that lay exactly on the unit sphere and thereby producing a random
-direction. This process of generating samples and rejecting them if they are not inside of a desired
-space is called _the rejection method_ and is found all over the literature. The method covered last
-chapter is referred to as _the inversion method_ because we invert a PDF.
+this vector produced points that lay exactly on the unit sphere and thereby represent a random
+direction. This process of generating samples and rejecting them if they are not inside a desired
+space is called _the rejection method_, and is found all over the literature. The method covered
+last chapter is referred to as _the inversion method_ because we invert a PDF.
 
 Every direction in 3D space has an associated point on the unit sphere and can be generated by
 solving for the vector that travels from the origin to that associated point. You can think of
@@ -1437,9 +1437,8 @@
 wavelength of the light: $\mathit{pScatter}(\omega_i, \omega_o, \lambda)$. A good example
 of this is a prism refracting white light into a rainbow. Lastly, the scattering PDF can also depend
 on the scattering position: $\mathit{pScatter}(\mathbf{x}, \omega_i, \omega_o, \lambda)$. The
-$\mathbf{x}$ is just a vector representing the scattering position: $\mathbf{x} = (x, y, z)$. The
-albedo of an object can also depend on these quantities:
-$A(\mathbf{x}, \omega_i, \omega_o, \lambda)$.
+$\mathbf{x}$ is just math notation for the scattering position: $\mathbf{x} = (x, y, z)$. The albedo
+of an object can also depend on these quantities: $A(\mathbf{x}, \omega_i, \omega_o, \lambda)$.
 
 <div class='together'>
 The color of a surface is found by integrating these terms over the unit hemisphere by the incident
@@ -1543,7 +1542,7 @@
 Playing with Importance Sampling
 ====================================================================================================
 <div class='together'>
-Our goal over the next four chapters is to instrument our program to send a bunch of extra rays
+Our goal over the next several chapters is to instrument our program to send a bunch of extra rays
 toward light sources so that our picture is less noisy. Let’s assume we can send a bunch of rays
 toward the light source using a PDF $\mathit{pLight}(\omega_o)$. Let’s also assume we have a PDF
 related to $\mathit{pScatter}$, and let’s call that $\mathit{pSurface}(\omega_o)$. A great thing
@@ -1599,8 +1598,8 @@
 </div>
 
 First, let’s instrument the code so that it explicitly samples some PDF and then normalizes for
-that. Remember Monte Carlo basics: $\int f(x) \approx \sum f(r)/p(r)$. For the Lambertian material, let’s
-sample like we do now: $p(\omega_o) = \cos(\theta_o) / \pi$.
+that. Remember Monte Carlo basics: $\int f(x) \approx \sum f(r)/p(r)$. For the Lambertian material,
+let’s sample like we do now: $p(\omega_o) = \cos(\theta_o) / \pi$.
 
 <div class='together'>
 We modify the base-class `material` to enable this importance sampling:
@@ -1702,7 +1701,7 @@
     The ray_color function, modified for importance sampling]
 </div>
 
-You should get exactly the same picture. Which, _should make sense_, as the scattered part of
+You should get exactly the same picture. Which _should make sense_, as the scattered part of
 `ray_color` is getting multiplied by `scattering_pdf / pdf`, and as `pdf` is equal to
 `scattering_pdf` is just the same as multiplying by one.
 
@@ -1788,13 +1787,13 @@
 attempt to repeat the uniform hemispherical scattering from the first book. There's nothing wrong
 with this technique, but we are no longer treating our objects as Lambertian. Lambertian is a
 specific type of diffuse material that requires a $\cos(\theta_o)$ scattering distribution.
-Uniform hemispherical scattering is another diffuse material, albeit a different one. If we keep the
-material the same but change the PDF, as we did in last section, we will still converge on the same
-answer, but our convergence may take more or less samples. However, if we change the material, we
-will have fundamentally changed the render and the algorithm will converge on a different answer. So
-when we replace Lambertian diffuse with uniform hemispherical diffuse we should expect that the
-outcome of our render to be _materially_ different. We're going to adjust our scattering direction
-and scattering PDF:
+Uniform hemispherical scattering is a different diffuse material. If we keep the material the same
+but change the PDF, as we did in last section, we will still converge on the same answer, but our
+convergence may take more or less samples. However, if we change the material, we will have
+fundamentally changed the render and the algorithm will converge on a different answer. So when we
+replace Lambertian diffuse with uniform hemispherical diffuse we should expect the outcome of our
+render to be _materially_ different. We're going to adjust our scattering direction and scattering
+PDF:
 
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
     class lambertian : public material {