diff --git a/books/RayTracingTheNextWeek.html b/books/RayTracingTheNextWeek.html
index 0bbc9704..5c7d5a69 100644
--- a/books/RayTracingTheNextWeek.html
+++ b/books/RayTracingTheNextWeek.html
@@ -2598,11 +2598,65 @@
$$ \mathbf{P} = \mathbf{Q} + \alpha \mathbf{u} + \beta \mathbf{v} $$
-If $\mathbf{u}$ and $\mathbf{v}$ were guaranteed to be orthogonal to each other (forming a 90° angle
-between them), then this would be a simple matter of using the dot product to project $\mathbf{P}$
-onto each of the basis vectors $\mathbf{u}$ and $\mathbf{v}$. However, since we are not restricting
+Pulling a rabbit out of my hat, the planar coordinates $\alpha$ and $\beta$ are given by the
+following equations:
+
+ $$ \alpha = \mathbf{w} \cdot (\mathbf{p} \times \mathbf{v}) $$
+ $$ \beta = \mathbf{w} \cdot (\mathbf{u} \times \mathbf{p}) $$
+
+where
+
+ $$ \mathbf{w} = \frac{\mathbf{n}}{\mathbf{n} \cdot (\mathbf{u} \times \mathbf{v})}
+ = \frac{\mathbf{n}}{\mathbf{n} \cdot \mathbf{n}}$$
+
+The vector $\mathbf{w}$ is constant for a given quadrilateral, so we'll cache that value.
+
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
+ class quad : public hittable {
+ public:
+ quad(const point3& _Q, const vec3& _u, const vec3& _v, shared_ptr m)
+ : Q(_Q), u(_u), v(_v), mat(m)
+ {
+ auto n = cross(u, v);
+ normal = unit_vector(n);
+ D = dot(normal, Q);
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
+ w = n / dot(n,n);
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
+
+ set_bounding_box();
+ }
+ ...
+
+ private:
+ point3 Q;
+ vec3 u, v;
+ shared_ptr mat;
+ aabb bbox;
+ vec3 normal;
+ double D;
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
+ vec3 w;
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
+ };
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ [Listing [quad-w]: [quad.h] Caching the quadrilateral's w value]
+
+
+Deriving the Planar Coordinates
+--------------------------------
+
+(This section covers the derivation of the equations above. Feel free to skip to the next section if
+you're not interested.)
+
+Refer back to figure [ray-plane]. If the planar basis vectors $\mathbf{u}$ and $\mathbf{v}$ were
+guaranteed to be orthogonal to each other (forming a 90° angle between them), then solving for
+$\alpha$ and $\beta$ would be a simple matter of using the dot product to project $\mathbf{P}$ onto
+each of the basis vectors $\mathbf{u}$ and $\mathbf{v}$. However, since we are not restricting
$\mathbf{u}$ and $\mathbf{v}$ to be orthogonal, the math's a little bit trickier.
+To set things up, consider that
+
$$ \mathbf{P} = \mathbf{Q} + \alpha \mathbf{u} + \beta \mathbf{v}$$
$$ \mathbf{p} = \mathbf{P} - \mathbf{Q} = \alpha \mathbf{u} + \beta \mathbf{v} $$
@@ -2610,7 +2664,7 @@
Here, $\mathbf{P}$ is the _point_ of intersection, and $\mathbf{p}$ is the _vector_ from
$\mathbf{Q}$ to $\mathbf{P}$.
-Cross the above equation with $\mathbf{u}$ and $\mathbf{v}$, respectively:
+Cross the equation for $\mathbf{p}$ with $\mathbf{u}$ and $\mathbf{v}$, respectively:
$$ \begin{align*}
\mathbf{u} \times \mathbf{p} &= \mathbf{u} \times (\alpha \mathbf{u} + \beta \mathbf{v}) \\
@@ -2668,39 +2722,6 @@
$$ \alpha = \mathbf{w} \cdot (\mathbf{p} \times \mathbf{v}) $$
$$ \beta = \mathbf{w} \cdot (\mathbf{u} \times \mathbf{p}) $$
-The vector $\mathbf{w}$ is constant for a given quadrilateral, so we'll cache that value.
-
- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
- class quad : public hittable {
- public:
- quad(const point3& _Q, const vec3& _u, const vec3& _v, shared_ptr m)
- : Q(_Q), u(_u), v(_v), mat(m)
- {
- auto n = cross(u, v);
- normal = unit_vector(n);
- D = dot(normal, Q);
- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
- w = n / dot(n,n);
- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
-
- set_bounding_box();
- }
- ...
-
- private:
- point3 Q;
- vec3 u, v;
- shared_ptr mat;
- aabb bbox;
- vec3 normal;
- double D;
- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
- vec3 w;
- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
- };
- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
- [Listing [quad-w]: [quad.h] Caching the quadrilateral's w value]
-
Interior Testing of The Intersection Using UV Coordinates
----------------------------------------------------------
@@ -2723,14 +2744,6 @@
That's the last piece needed to implement quadrilateral primitives.
-Pause a bit here and consider that if you use the $(\alpha,\beta)$ coordinates to determine if a
-point lies inside a quadrilateral (parallelogram), it's not too hard to imagine using these same 2D
-coordinates to determine if the intersection point lies inside _any_ other 2D (planar) primitive!
-
-We'll leave these additional 2D shape possibilities as an exercise to the reader, depending on your
-desire to explore. Consider triangles, disks, and rings (all of these are surprisingly easy). You
-could even create cut-out stencils based on the pixels of a texture map, or a Mandelbrot shape!
-
In order to make such experimentation a bit easier, we'll factor out the $(\alpha,\beta)$ interior
test method from the hit method.
@@ -2863,6 +2876,23 @@
+Additional 2D Primitives
+-------------------------
+Pause a bit here and consider that if you use the $(\alpha,\beta)$ coordinates to determine if a
+point lies inside a quadrilateral (parallelogram), it's not too hard to imagine using these same 2D
+coordinates to determine if the intersection point lies inside _any_ other 2D (planar) primitive!
+
+For example, suppose we change the `is_interior()` function to return true if `sqrt(a*a + b*b) < r`.
+This would then implement disk primitives of radius `r`. For triangles, try
+`a > 0 && b > 0 && a + b < 1`.
+
+We'll leave additional 2D shape possibilities as an exercise to the reader, depending on your desire
+to explore. You could even create cut-out stencils based on the pixels of a texture map, or a
+Mandelbrot shape! As a little Easter egg, check out the `alternate-2D-primitves` tag in the source
+repository. This has solutions for triangles, ellipses and annuli (rings) in
+`src/TheNextWeek/quad.h`
+
+
Lights
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