Implement bubbles with IOR, not negative radii

We were using negative radii on spheres to invert the sense of front vs
back faces of spheres, and using this to get an inverted index of
refraction (IOR) for bubbles. This method was used to model a hollow
glass sphere.

Over the past couple of years we've encountered a number of bugs induced
by negative radii spheres (see the list in the description for issue
1420).

Instead, this change interprets the dielectric parameter not as an IOR
relative to surrounding air, but instead as the ratio of the IOR of the
object material over the IOR of the surrounding medium. Interpreting the
parameter this way lets you model a sphere of air enclosed in glass --
just what we need to model a hollow glass sphere purely with the proper
IORs, and without resorting to inverted geometry with negative radii
spheres.

We also end up using this method to properly demonstrate total
internal/external reflection in another section.

Resolves #1420

Author: hollasch

books/RayTracingInOneWeekend.html

@@ -1114,7 +1114,7 @@
 
     class sphere : public hittable {
       public:
-        sphere(const point3& center, double radius) : center(center), radius(radius) {}
+        sphere(const point3& center, double radius) : center(center), radius(fmax(0,radius)) {}
 
         bool hit(const ray& r, double ray_tmin, double ray_tmax, hit_record& rec) const override {
             vec3 oc = center - r.origin();
@@ -2795,7 +2795,7 @@
       public:
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
         sphere(const point3& center, double radius, shared_ptr<material> mat)
-          : center(center), radius(radius), mat(mat) {}
+          : center(center), radius(fmax(0,radius)), mat(mat) {}
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
 
         bool hit(const ray& r, interval ray_t, hit_record& rec) const override {
@@ -3477,50 +3477,47 @@
 
 Modeling a Hollow Glass Sphere
 -------------------------------
-An interesting and easy trick with dielectric spheres is to note that if you use a negative radius,
-the geometry is unaffected, but the surface normal points inward.
+Let's model a hollow glass sphere. This is a sphere of some thickness with another sphere of air
+inside it. If you think about the path of a ray going through such an object, it will hit the outer
+sphere, refract, hit the inner sphere (assuming we do hit it), refract a second time, and travel
+through the air inside. Then it will continue on, hit the inside surface of the inner sphere,
+refract back, then hit the inside surface of the outer sphere, and finally refract and exit back
+into the scene atmosphere.
 
-However, properly handling negative radii can be tricky. Recall the line from `sphere::hit()` in
-listing [sphere-material] that calculates the outward normal:
+The outer sphere is just modeled with a standard glass sphere, with an index of refraction of around
+1.50. The inner sphere is a bit different, because _its_ index of refraction will now be relative to
+the "glass atmosphere" of the outer sphere. This is actually simple to specify, as our
+index-of-refraction parameter to the dielectric material can be interpreted as the ratio of the
+index of refraction of the object over the index of refraction of the enclosing medium.
 
-    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
-    vec3 outward_normal = (rec.p - center) / radius;
-    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-    [Listing [proper-invert-sphere-normal]: Proper normal handling for spheres with negative radii]
-
-In your own implementation, you might have been tempted to instead do something like this:
-
-    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
-    vec3 outward_normal = (rec.p - center).unit_vector();
-    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-    [Listing [improper-invert-sphere-normal]:
-        Problematic normal calculation for spheres with negative radii
-    ]
-
-If you do that, spheres with negative radii won't work properly. Since a sphere with a negative
-radius is a _bubble_, its interior is the infinite space outside the sphere. Its exterior is the
-finite bubble inside the sphere, so the outward normal needs to point toward the sphere center.
-Dividing by the (negative) radius flips the normal as we want. If you implmented your code like the
-second example above, you'll want to fix that now.
+In this case, the inner sphere would have an index of refraction of air (the inner sphere material)
+over the index of refraction of glass (the enclosing medium), or $1.00/1.50 = 0.67$.
 
-Let's use this hollow sphere hack to model the interior of a sphere with a given thickness. To do
-this, add a second _inverted_ glass sphere inside the original glass sphere:
+Here's the code:
 
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
     ...
+    auto material_ground = make_shared<lambertian>(color(0.8, 0.8, 0.0));
+    auto material_center = make_shared<lambertian>(color(0.1, 0.2, 0.5));
+    auto material_left   = make_shared<dielectric>(1.50);
+    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
+    auto material_bubble = make_shared<dielectric>(0.67);
+    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
+    auto material_right  = make_shared<metal>(color(0.8, 0.6, 0.2), 0.0);
+
     world.add(make_shared<sphere>(point3( 0.0, -100.5, -1.0), 100.0, material_ground));
     world.add(make_shared<sphere>(point3( 0.0,    0.0, -1.0),   0.5, material_center));
     world.add(make_shared<sphere>(point3(-1.0,    0.0, -1.0),   0.5, material_left));
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
-    world.add(make_shared<sphere>(point3(-1.0,    0.0, -1.0),  -0.4, material_left));
+    world.add(make_shared<sphere>(point3(-1.0,    0.0, -1.0),   0.4, material_bubble));
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
     world.add(make_shared<sphere>(point3( 1.0,    0.0, -1.0),   0.5, material_right));
     ...
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
     [Listing [scene-hollow-glass]: <kbd>[main.cc]</kbd> Scene with hollow glass sphere]
 
 <div class='together'>
-This gives:
+And here's the result:
 
   ![<span class='num'>Image 18:</span> A hollow glass sphere
   ](../images/img-1.18-glass-hollow.png class='pixel')
@@ -3772,13 +3769,14 @@
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
         auto material_ground = make_shared<lambertian>(color(0.8, 0.8, 0.0));
         auto material_center = make_shared<lambertian>(color(0.1, 0.2, 0.5));
-        auto material_left   = make_shared<dielectric>(1.5);
+        auto material_left   = make_shared<dielectric>(1.50);
+        auto material_bubble = make_shared<dielectric>(0.67);
         auto material_right  = make_shared<metal>(color(0.8, 0.6, 0.2), 0.0);
 
         world.add(make_shared<sphere>(point3( 0.0, -100.5, -1.0), 100.0, material_ground));
         world.add(make_shared<sphere>(point3( 0.0,    0.0, -1.0),   0.5, material_center));
         world.add(make_shared<sphere>(point3(-1.0,    0.0, -1.0),   0.5, material_left));
-        world.add(make_shared<sphere>(point3(-1.0,    0.0, -1.0),  -0.4, material_left));
+        world.add(make_shared<sphere>(point3(-1.0,    0.0, -1.0),   0.4, material_bubble));
         world.add(make_shared<sphere>(point3( 1.0,    0.0, -1.0),   0.5, material_right));
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
 

books/RayTracingTheNextWeek.html

@@ -179,12 +179,12 @@
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
         // Stationary Sphere
         sphere(const point3& center, double radius, shared_ptr<material> mat)
-          : center1(center), radius(radius), mat(mat), is_moving(false) {}
+          : center1(center), radius(fmax(0,radius)), mat(mat), is_moving(false) {}
 
         // Moving Sphere
         sphere(const point3& center1, const point3& center2, double radius,
                shared_ptr<material> mat)
-          : center1(center1), radius(radius), mat(mat), is_moving(true)
+          : center1(center1), radius(fmax(0,radius)), mat(mat), is_moving(true)
         {
             center_vec = center2 - center1;
         }
@@ -724,7 +724,7 @@
         // Stationary Sphere
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
         sphere(const point3& center, double radius, shared_ptr<material> mat)
-          : center1(center), radius(radius), mat(mat), is_moving(false)
+          : center1(center), radius(fmax(0,radius)), mat(mat), is_moving(false)
         {
             auto rvec = vec3(radius, radius, radius);
             bbox = aabb(center1 - rvec, center1 + rvec);
@@ -760,7 +760,7 @@
         // Moving Sphere
         sphere(const point3& center1, const point3& center2, double radius,
                shared_ptr<material> mat)
-          : center1(center1), radius(radius), mat(mat), is_moving(true)
+          : center1(center1), radius(fmax(0,radius)), mat(mat), is_moving(true)
         {
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
             auto rvec = vec3(radius, radius, radius);

src/InOneWeekend/sphere.h

@@ -19,7 +19,7 @@
 class sphere : public hittable {
   public:
     sphere(const point3& center, double radius, shared_ptr<material> mat)
-      : center(center), radius(radius), mat(mat) {}
+      : center(center), radius(fmax(0,radius)), mat(mat) {}
 
     bool hit(const ray& r, interval ray_t, hit_record& rec) const override {
         vec3 oc = center - r.origin();

src/TheNextWeek/sphere.h

@@ -20,7 +20,7 @@ class sphere : public hittable {
   public:
     // Stationary Sphere
     sphere(const point3& center, double radius, shared_ptr<material> mat)
-      : center1(center), radius(radius), mat(mat), is_moving(false)
+      : center1(center), radius(fmax(0,radius)), mat(mat), is_moving(false)
     {
         auto rvec = vec3(radius, radius, radius);
         bbox = aabb(center1 - rvec, center1 + rvec);
@@ -29,7 +29,7 @@ class sphere : public hittable {
     // Moving Sphere
     sphere(const point3& center1, const point3& center2, double radius,
            shared_ptr<material> mat)
-      : center1(center1), radius(radius), mat(mat), is_moving(true)
+      : center1(center1), radius(fmax(0,radius)), mat(mat), is_moving(true)
     {
         auto rvec = vec3(radius, radius, radius);
         aabb box1(center1 - rvec, center1 + rvec);

src/TheRestOfYourLife/sphere.h

@@ -21,7 +21,7 @@ class sphere : public hittable {
   public:
     // Stationary Sphere
     sphere(const point3& center, double radius, shared_ptr<material> mat)
-      : center1(center), radius(radius), mat(mat), is_moving(false)
+      : center1(center), radius(fmax(0,radius)), mat(mat), is_moving(false)
     {
         auto rvec = vec3(radius, radius, radius);
         bbox = aabb(center1 - rvec, center1 + rvec);
@@ -30,7 +30,7 @@ class sphere : public hittable {
     // Moving Sphere
     sphere(const point3& center1, const point3& center2, double radius,
            shared_ptr<material> mat)
-      : center1(center1), radius(radius), mat(mat), is_moving(true)
+      : center1(center1), radius(fmax(0,radius)), mat(mat), is_moving(true)
     {
         auto rvec = vec3(radius, radius, radius);
         aabb box1(center1 - rvec, center1 + rvec);