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 {
@@ -3157,6 +3157,25 @@
 splits into a reflected ray and a refracted (transmitted) ray. We’ll handle that by randomly
 choosing between reflection and refraction, only generating one scattered ray per interaction.
 
+As a quick review of terms, a _reflected_ ray hits a surface and then "bounces" off in a new
+direction.
+
+A _refracted_ ray bends as it transitions from a material's surroundings into the material itself
+(as with glass or water). This is why a pencil looks bent when partially inserted in water.
+
+The amount that a refracted ray bends is determined by the material's _refractive index_. Generally,
+this is a single value that describes how much light bends when entering a material from a vacuum.
+Glass has a refractive index of something like 1.5&ndash;1.7, and air has a small refractive index
+of 1.000293.
+
+When a transparent material is embedded in a different transparent material, you can describe the
+refraction with a relative refraction index: the refractive index of the object's material divided
+by the refractive index of the surrounding material. For example, if you want to render a glass ball
+under water, then the glass ball would have an effective refractive index of 1.125. This is given by
+the refractive index of glass (1.5) divided by the refractive index of water (1.333).
+
+You can find the refractive index of most common materials with a quick internet search.
+
 
 Refraction
 -----------
@@ -3257,23 +3276,24 @@
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ Highlight
     class dielectric : public material {
       public:
-        dielectric(double ref_index) : ref_index(ref_index) {}
+        dielectric(double refraction_index) : refraction_index(refraction_index) {}
 
         bool scatter(const ray& r_in, const hit_record& rec, color& attenuation, ray& scattered)
         const override {
             attenuation = color(1.0, 1.0, 1.0);
-            double refraction_ratio = rec.front_face ? (1.0/ref_index) : ref_index;
+            double ri = rec.front_face ? (1.0/refraction_index) : refraction_index;
 
             vec3 unit_direction = unit_vector(r_in.direction());
-            vec3 refracted = refract(unit_direction, rec.normal, refraction_ratio);
+            vec3 refracted = refract(unit_direction, rec.normal, ri);
 
             scattered = ray(rec.p, refracted);
             return true;
         }
 
       private:
-        double ref_index;  // Refractive index in vacuum or air, or the ratio of the material's
-                           // refractive index over the refractive index of the enclosing media
+        // Refractive index in vacuum or air, or the ratio of the material's refractive index over
+        // the refractive index of the enclosing media
+        double refraction_index;
     };
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
     [Listing [dielectric-always-refract]: <kbd>[material.h]</kbd>
@@ -3327,7 +3347,7 @@
 solution does not exist, the glass cannot refract, and therefore must reflect the ray:
 
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
-    if (refraction_ratio * sin_theta > 1.0) {
+    if (ri * sin_theta > 1.0) {
         // Must Reflect
         ...
     } else {
@@ -3355,7 +3375,7 @@
     double cos_theta = fmin(dot(-unit_direction, rec.normal), 1.0);
     double sin_theta = sqrt(1.0 - cos_theta*cos_theta);
 
-    if (refraction_ratio * sin_theta > 1.0) {
+    if (ri * sin_theta > 1.0) {
         // Must Reflect
         ...
     } else {
@@ -3371,34 +3391,35 @@
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
     class dielectric : public material {
       public:
-        dielectric(double ref_index) : ref_index(ref_index) {}
+        dielectric(double refraction_index) : refraction_index(refraction_index) {}
 
         bool scatter(const ray& r_in, const hit_record& rec, color& attenuation, ray& scattered)
         const override {
             attenuation = color(1.0, 1.0, 1.0);
-            double refraction_ratio = rec.front_face ? (1.0/ref_index) : ref_index;
+            double ri = rec.front_face ? (1.0/refraction_index) : refraction_index;
 
             vec3 unit_direction = unit_vector(r_in.direction());
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
             double cos_theta = fmin(dot(-unit_direction, rec.normal), 1.0);
             double sin_theta = sqrt(1.0 - cos_theta*cos_theta);
 
-            bool cannot_refract = refraction_ratio * sin_theta > 1.0;
+            bool cannot_refract = ri * sin_theta > 1.0;
             vec3 direction;
 
             if (cannot_refract)
                 direction = reflect(unit_direction, rec.normal);
             else
-                direction = refract(unit_direction, rec.normal, refraction_ratio);
+                direction = refract(unit_direction, rec.normal, ri);
 
             scattered = ray(rec.p, direction);
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
             return true;
         }
 
       private:
-        double ref_index;  // Refractive index in vacuum or air, or the ratio of the material's
-                           // refractive index over the refractive index of the enclosing media
+        // Refractive index in vacuum or air, or the ratio of the material's refractive index over
+        // the refractive index of the enclosing media
+        double refraction_index;
     };
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
     [Listing [dielectric-with-refraction]: <kbd>[material.h]</kbd>
@@ -3440,40 +3461,41 @@
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
     class dielectric : public material {
       public:
-        dielectric(double ref_index) : ref_index(ref_index) {}
+        dielectric(double refraction_index) : refraction_index(refraction_index) {}
 
         bool scatter(const ray& r_in, const hit_record& rec, color& attenuation, ray& scattered)
         const override {
             attenuation = color(1.0, 1.0, 1.0);
-            double refraction_ratio = rec.front_face ? (1.0/ref_index) : ref_index;
+            double ri = rec.front_face ? (1.0/refraction_index) : refraction_index;
 
             vec3 unit_direction = unit_vector(r_in.direction());
             double cos_theta = fmin(dot(-unit_direction, rec.normal), 1.0);
             double sin_theta = sqrt(1.0 - cos_theta*cos_theta);
 
-            bool cannot_refract = refraction_ratio * sin_theta > 1.0;
+            bool cannot_refract = ri * sin_theta > 1.0;
             vec3 direction;
 
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
-            if (cannot_refract || reflectance(cos_theta, refraction_ratio) > random_double())
+            if (cannot_refract || reflectance(cos_theta, ri) > random_double())
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
                 direction = reflect(unit_direction, rec.normal);
             else
-                direction = refract(unit_direction, rec.normal, refraction_ratio);
+                direction = refract(unit_direction, rec.normal, ri);
 
             scattered = ray(rec.p, direction);
             return true;
         }
 
       private:
-        double ref_index;  // Refractive index in vacuum or air, or the ratio of the material's
-                           // refractive index over the refractive index of the enclosing media
+        // Refractive index in vacuum or air, or the ratio of the material's refractive index over
+        // the refractive index of the enclosing media
+        double refraction_index;
 
 
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
-        static double reflectance(double cosine, double ref_idx) {
+        static double reflectance(double cosine, double refraction_index) {
             // Use Schlick's approximation for reflectance.
-            auto r0 = (1-ref_idx) / (1+ref_idx);
+            auto r0 = (1-refraction_index) / (1+refraction_index);
             r0 = r0*r0;
             return r0 + (1-r0)*pow((1 - cosine),5);
         }
@@ -3485,50 +3507,49 @@
 
 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.
-
-However, properly handling negative radii can be tricky. Recall the line from `sphere::hit()` in
-listing [sphere-material] that calculates the outward normal:
-
-    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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:
+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.
 
-    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
-    vec3 outward_normal = (rec.p - center).unit_vector();
-    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-    [Listing [improper-invert-sphere-normal]:
-        Problematic normal calculation for spheres with negative radii
-    ]
+The outer sphere is just modeled with a standard glass sphere, with a refractive index of around
+1.50 (modeling a refraction from the outside air into glass). The inner sphere is a bit different
+because _its_ refractive index should be relative to the material of the surrounding outer sphere,
+thus modeling a transition from glass into the inner air.
 
-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.
+This is actually simple to specify, as the `refraction_index` parameter to the dielectric material
+can be interpreted as the _ratio_ of the refractive index of the object divided by the refractive
+index of the enclosing medium. In this case, the inner sphere would have an refractive index 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')
@@ -3780,13 +3801,14 @@
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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);

books/RayTracingTheRestOfYourLife.html

@@ -3316,7 +3316,7 @@
 
     class dielectric : public material {
       public:
-        dielectric(double ref_index) : ref_index(ref_index) {}
+        dielectric(double refraction_index) : refraction_index(refraction_index) {}
 
 
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
@@ -3325,19 +3325,19 @@
             srec.pdf_ptr = nullptr;
             srec.skip_pdf = true;
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
-            double refraction_ratio = rec.front_face ? (1.0/ref_index) : ref_index;
+            double ri = rec.front_face ? (1.0/refraction_index) : refraction_index;
 
             vec3 unit_direction = unit_vector(r_in.direction());
             double cos_theta = fmin(dot(-unit_direction, rec.normal), 1.0);
             double sin_theta = sqrt(1.0 - cos_theta*cos_theta);
 
-            bool cannot_refract = refraction_ratio * sin_theta > 1.0;
+            bool cannot_refract = ri * sin_theta > 1.0;
             vec3 direction;
 
-            if (cannot_refract || reflectance(cos_theta, refraction_ratio) > random_double())
+            if (cannot_refract || reflectance(cos_theta, ri) > random_double())
                 direction = reflect(unit_direction, rec.normal);
             else
-                direction = refract(unit_direction, rec.normal, refraction_ratio);
+                direction = refract(unit_direction, rec.normal, ri);
 
 
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight