Fix line lengths in Book3

Author: ymherklotz

books/RayTracingTheRestOfYourLife.html

@@ -1340,14 +1340,14 @@
 in the _LMS color space_ (long, medium, short).
 
 If the light does scatter, it will have a directional distribution that we can describe as a PDF
-over solid angle. I will refer to this as its _scattering PDF_: $\mathit{pScatter}()$. The scattering PDF
-will vary with the outgoing direction: $\mathit{pScatter}(\omega_o)$. The scattering PDF can also vary with
-_incident direction_: $\mathit{pScatter}(\omega_i, \omega_o)$. You can see this varying with incident
-direction when you look at reflections off a road -- they become mirror-like as your viewing angle
-(incident angle) approaches grazing. Lastly, the scattering PDF can also depend on the scattering
-position: $\mathit{pScatter}(\textbf{x}, \omega_i, \omega_o)$. The $\textbf{x}$ is just a vector representing
-the scattering position: $\textbf{x} = (x, y, z)$. The Albedo of an object can also depend on these
-quantities: $A(\textbf{x}, \omega_i, \omega_o)$.
+over solid angle. I will refer to this as its _scattering PDF_: $\mathit{pScatter}()$. The
+scattering PDF will vary with the outgoing direction: $\mathit{pScatter}(\omega_o)$. The scattering
+PDF can also vary with _incident direction_: $\mathit{pScatter}(\omega_i, \omega_o)$. You can see
+this varying with incident direction when you look at reflections off a road -- they become
+mirror-like as your viewing angle (incident angle) approaches grazing. Lastly, the scattering PDF
+can also depend on the scattering position: $\mathit{pScatter}(\textbf{x}, \omega_i, \omega_o)$. The
+$\textbf{x}$ is just a vector representing the scattering position: $\textbf{x} = (x, y, z)$. The
+Albedo of an object can also depend on these quantities: $A(\textbf{x}, \omega_i, \omega_o)$.
 
 <div class='together'>
 The color of a surface is found by integrating these terms over the unit hemisphere by the incident
@@ -1363,8 +1363,8 @@
 that is shining on that point. This is a recursive algorithm, and is the reason our `ray_color`
 function returns the color of the current object multiplied by the color of the next ray.
 
-Note that $A()$, $\mathit{pScatter}()$, and $\text{Color}_i()$ may all depend on the wavelength of the light,
-$\lambda$, but I've left it out of the equation because it's complicated enough as it is.
+Note that $A()$, $\mathit{pScatter}()$, and $\text{Color}_i()$ may all depend on the wavelength of
+the light, $\lambda$, but I've left it out of the equation because it's complicated enough as it is.
 </div>
 
 
@@ -1385,9 +1385,9 @@
 For a Lambertian surface we already implicitly implemented this formula for the special case where
 $p()$ is a cosine density. The $\mathit{pScatter}()$ of a Lambertian surface is proportional to
 $\cos(\theta_o)$, where $\theta_o$ is the angle relative to the surface normal. All two dimensional
-PDFs need to integrate to one over the whole surface (remember that $\mathit{pScatter}()$ is a PDF). We set
-$\mathit{pScatter}(\theta_o < 0) = 0$ so that we don't scatter below the horizon. The integral of
-$\cos(\theta_o)$ over the hemisphere is $\pi$.
+PDFs need to integrate to one over the whole surface (remember that $\mathit{pScatter}()$ is a
+PDF). We set $\mathit{pScatter}(\theta_o < 0) = 0$ so that we don't scatter below the horizon. The
+integral of $\cos(\theta_o)$ over the hemisphere is $\pi$.
 
 <div class='together'>
 To integrate over the hemisphere, remember that in spherical coordinates:
@@ -1446,27 +1446,29 @@
 <div class='together'>
 Our goal over the next two 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 about PDFs is that you can just
-use linear mixtures of them to form mixture densities that are also PDFs. For example, the simplest
-would be:
+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
+about PDFs is that you can just use linear mixtures of them to form mixture densities that are also
+PDFs. For example, the simplest would be:
 
-  $$ p(\omega_o) = \frac{1}{2} \mathit{pSurface}(\omega_o) +  \frac{1}{2} \mathit{pLight}(\omega_o)$$
+  $$ p(\omega_o) = \frac{1}{2} \mathit{pSurface}(\omega_o) +  \frac{1}{2}
+      \mathit{pLight}(\omega_o)$$
 </div>
 
 As long as the weights are positive and add up to one, any such mixture of PDFs is a PDF. Remember,
 we can use any PDF: _all PDFs eventually converge to the correct answer_. So, the game is to figure
 out how to make the PDF larger where the product
 
-    $$ \mathit{pScatter}(\textbf{x}, \omega_i, \omega_o) \cdot \text{Color}_i(\textbf{x}, \omega_i) $$
+    $$ \mathit{pScatter}(\textbf{x}, \omega_i, \omega_o) \cdot
+        \text{Color}_i(\textbf{x}, \omega_i) $$
 
 is largest. For diffuse surfaces, this is mainly a matter of guessing where
 $\text{Color}_i(\textbf{x}, \omega_i)$ is largest. Which is equivalent to guessing where the most
 light is coming from.
 
-For a mirror, $\mathit{pScatter}()$ is huge only near one direction, so $\mathit{pScatter}()$ matters a lot more. In
-fact, most renderers just make mirrors a special case, and make the $\mathit{pScatter}()/p()$ implicit -- our
-code currently does that.
+For a mirror, $\mathit{pScatter}()$ is huge only near one direction, so $\mathit{pScatter}()$
+matters a lot more. In fact, most renderers just make mirrors a special case, and make the
+$\mathit{pScatter}()/p()$ implicit -- our code currently does that.
 
 
 Returning to the Cornell Box
@@ -2416,9 +2418,10 @@
 </div>
 
 <div class='together'>
-The details of how this is done under the hood varies for $\mathit{pSurface}$ and $\mathit{pLight}$, but that is
-exactly what class hierarchies were invented for! It’s never obvious what goes in an abstract class,
-so my approach is to be greedy and hope a minimal interface works, and for `pdf` this implies:
+The details of how this is done under the hood varies for $\mathit{pSurface}$ and $\mathit{pLight}$,
+but that is exactly what class hierarchies were invented for! It’s never obvious what goes in an
+abstract class, so my approach is to be greedy and hope a minimal interface works, and for `pdf`
+this implies:
 
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
     #ifndef PDF_H
@@ -2741,7 +2744,8 @@
 
 For example, we could just average the two densities:
 
-  $$ \mathit{pMixture}(\omega_o) = \frac{1}{2} \mathit{pSurface}(\omega_o) +  \frac{1}{2} \mathit{pLight}(\omega_o)$$
+  $$ \mathit{pMixture}(\omega_o) = \frac{1}{2} \mathit{pSurface}(\omega_o) +  \frac{1}{2}
+      \mathit{pLight}(\omega_o)$$
 
 <div class='together'>
 How would we instrument our code to do that? There is a very important detail that makes this not
@@ -2769,10 +2773,10 @@
 correct, we would have to solve backwards to figure which PDF the direction could come from. Which
 honestly sounds like a nightmare, but fortunately we don't need to do that. There are some
 directions that both PDFs could have generated. For example, a direction toward the light could have
-been generated by either $\mathit{pLight}$ _or_ $\mathit{pSurface}$. It is sufficient for us to solve for the pdf
-value of $\mathit{pSurface}$ and of $\mathit{pLight}$ for a random direction and then take the PDF mixture weights to
-solve for the total PDF value for that direction. The mixture density class is actually pretty
-straightforward:
+been generated by either $\mathit{pLight}$ _or_ $\mathit{pSurface}$. It is sufficient for us to
+solve for the pdf value of $\mathit{pSurface}$ and of $\mathit{pLight}$ for a random direction and
+then take the PDF mixture weights to solve for the total PDF value for that direction. The mixture
+density class is actually pretty straightforward:
 
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
     class mixture_pdf : public pdf {