T4 Refraction (Milestone 2)

T4.1 Add refract() to calculate the refrac vector direction (vec3.h)

Add refract() in vec3.h to calculate the refrac vector direction

Add refract() at the end of vec3.h, outside the vec3 class.

inline vec3 refract(const vec3& uv, const vec3& n, double etai_over_etat) {
    auto cos_theta = std::fmin(dot(-uv, n), 1.0);
    vec3 r_out_perp =  etai_over_etat * (uv + cos_theta*n);
    vec3 r_out_parallel = -std::sqrt(std::fabs(1.0 - r_out_perp.length_squared())) * n;
    return r_out_perp + r_out_parallel;
}

T4.2 Add dielectric material (material.h)

Add class dielectric in material.h

dielectric inherits from material and implements its scatter method using refract()

class material {
//...
};

class lambertian : public material {
//...
};

class metal : public material {
//...
};

//>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
class dielectric : public material {
  private:
    // 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;

  public:
    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 ri = rec.front_face ? (1.0/refraction_index) : refraction_index;

        vec3 unit_direction = unit_vector(r_in.direction());
        // refract vector
        vec3 refracted = refract(unit_direction, rec.normal, ri);

        scattered = ray(rec.p, refracted);
        return true;
    }
};
//<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<

T4.3 Update the scene (main.cpp)

Change material left from metal to dielectric in main()

void main() {
    hittable_list world;

    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); // <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
    auto material_right  = make_shared<metal>(color(0.8, 0.6, 0.2));
    
    //...
}

Build and Run Your Program

If building is successful, run your program in the terminal as follows using your executable name

 ./ray01.exe > image11.ppm

It is slower then reflection as more refracted rays are involved.

If successful, you are going to see a lovely photo like the following

It is very interesting to notice that you see the sky colour through the bottom half of the glass sphere due to refraction.

T4.4 Add total internal reflection and Schlick Approximation in

Revise the satter() method in class dielectric to add total internal reflection, and schlick approximation for Fresnel effect.

class dielectric : public material {
  private:
    // 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;

    //>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
    static double reflectance(double cosine, double refraction_index) {
        // Use Schlick's approximation for reflectance.
        auto r0 = (1 - refraction_index) / (1 + refraction_index);
        r0 = r0*r0;
        return r0 + (1-r0)*std::pow((1 - cosine),5);
    }
    //<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<

  public:
    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 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, ri);

        // >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
        double cos_theta = std::fmin(dot(-unit_direction, rec.normal), 1.0);
        double sin_theta = std::sqrt(1.0 - cos_theta*cos_theta);

        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, ri);

        if (cannot_refract || reflectance(cos_theta, ri) > random_double())
            direction = reflect(unit_direction, rec.normal);
        else
            direction = refract(unit_direction, rec.normal, ri);
        
        scattered = ray(rec.p, direction);
        // <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<

        
        return true;
    }
};

Add new material in main() and add a hollow glass sphere

    void main() {
    // ...
    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); 
    auto material_bubble = make_shared<dielectric>(1.00 / 1.50); // <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
    auto material_right  = make_shared<metal>(color(0.8, 0.6, 0.2));

    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, -1.2),   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_bubble)); 
    // <<<<<<<<<<<<<<<
    world.add(make_shared<sphere>(point3( 1.0,    0.0, -1.0),   0.5, material_right));
    
    // ...

You are going to see output similar to the following: