Update books: refactored dielectric::scatter()

- Simpler reflection/refraction branching
- Global `schlick` function moved to dielectric private `reflectance`
  function.

Author: hollasch

books/RayTracingInOneWeekend.html

@@ -2405,6 +2405,7 @@
 
                 vec3 unit_direction = unit_vector(r_in.direction());
                 vec3 refracted = refract(unit_direction, rec.normal, refraction_ratio);
+
                 scattered = ray(rec.p, refracted);
                 return true;
             }
@@ -2520,17 +2521,15 @@
                 double sin_theta = sqrt(1.0 - cos_theta*cos_theta);
 
                 bool cannot_refract = refraction_ratio * sin_theta > 1.0;
+                vec3 direction;
 
-                // If the ray cannot refract, then return the reflected path.
                 if (cannot_refract) {
-                    vec3 reflected = reflect(unit_direction, rec.normal);
-                    scattered = ray(rec.p, reflected);
-                    return true;
-                }
-    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
+                    direction = reflect(unit_direction, rec.normal);
+                else
+                    direction = refract(unit_direction, rec.normal, refraction_ratio);
 
-                vec3 refracted = refract(unit_direction, rec.normal, refraction_ratio);
-                scattered = ray(rec.p, refracted);
+                scattered = ray(rec.p, direction);
+    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
                 return true;
             }
 
@@ -2566,20 +2565,9 @@
 <div class='together'>
 Now real glass has reflectivity that varies with angle -- look at a window at a steep angle and it
 becomes a mirror. There is a big ugly equation for that, but almost everybody uses a cheap and
-surprisingly accurate polynomial approximation by Christophe Schlick:
-
-    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
-    double schlick(double cosine, double ref_idx) {
-        auto r0 = (1-ref_idx) / (1+ref_idx);
-        r0 = r0*r0;
-        return r0 + (1-r0)*pow((1 - cosine),5);
-    }
-    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-    [Listing [schlick]: <kbd>[material.h]</kbd> Schlick approximation]
-</div>
+surprisingly accurate polynomial approximation by Christophe Schlick. This yields our full glass
+material:
 
-<div class='together'>
-This yields our full glass material:
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
     class dielectric : public material {
         public:
@@ -2596,24 +2584,31 @@
                 double sin_theta = sqrt(1.0 - cos_theta*cos_theta);
 
                 bool cannot_refract = refraction_ratio * sin_theta > 1.0;
+                vec3 direction;
 
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
-                // If the ray cannot refract, or if it probabilistically reflects because of its
-                // grazing angle, then return the reflected path.
-                if (cannot_refract || random_double() < schlick(cos_theta, refraction_ratio)) {
+                if (cannot_refract || reflectance(cos_theta, refraction_ratio) > random_double())
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
-                    vec3 reflected = reflect(unit_direction, rec.normal);
-                    scattered = ray(rec.p, reflected);
-                    return true;
-                }
+                    direction = reflect(unit_direction, rec.normal);
+                else
+                    direction = refract(unit_direction, rec.normal, refraction_ratio);
 
-                vec3 refracted = refract(unit_direction, rec.normal, refraction_ratio);
-                scattered = ray(rec.p, refracted);
+                scattered = ray(rec.p, direction);
                 return true;
             }
 
         public:
             double ir; // Index of Refraction
+    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
+
+        private:
+            static double reflectance(double cosine, double ref_idx) {
+                // Use Schlick's approximation for reflectance.
+                auto r0 = (1-ref_idx) / (1+ref_idx);
+                r0 = r0*r0;
+                return r0 + (1-r0)*pow((1 - cosine),5);
+            }
+    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
     };
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
     [Listing [glass]: <kbd>[material.h]</kbd> Full glass material]

books/RayTracingTheNextWeek.html

@@ -308,21 +308,9 @@
     class dielectric : public material {
         ...
             virtual bool scatter(
-                const ray& r_in, const hit_record& rec, color& attenuation, ray& scattered
-            ) const override {
-
                 ...
-                if (cannot_refract || random_double() < schlick(cos_theta, refraction_ratio)) {
-                    vec3 reflected = reflect(unit_direction, rec.normal);
-    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
-                    scattered = ray(rec.p, reflected, r_in.time());
-    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
-                    return true;
-                }
-
-                vec3 refracted = refract(unit_direction, rec.normal, etai_over_etat);
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
-                scattered = ray(rec.p, refracted, r_in.time());
+                scattered = ray(rec.p, direction, r_in.time());
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
                 return true;
             }

books/RayTracingTheRestOfYourLife.html

@@ -2161,19 +2161,9 @@
                 srec.attenuation = color(1.0, 1.0, 1.0);
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
                 double refraction_ratio = rec.front_face ? (1.0/ir) : ir;
-
                 ...
-                if (cannot_refract || random_double() < schlick(cos_theta, refraction_ratio)) {
-                    vec3 reflected = reflect(unit_direction, rec.normal);
-    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
-                    srec.specular_ray = ray(rec.p, reflected, r_in.time());
-    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
-                    return true;
-                }
-
-                vec3 refracted = refract(unit_direction, rec.normal, refraction_ratio);
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++ highlight
-                srec.specular_ray = ray(rec.p, refracted, r_in.time());
+                srec.specular_ray = ray(rec.p, direction, r_in.time());
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
                 return true;
             }