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**Chapter 9: Dielectrics**
Clear materials such as water, glass, and diamonds are dielectrics. When a light ray hits them, it
splits into a reflected ray and a refracted (transmitted) ray. We’ll handle that by randomly
choosing between reflection or refraction and only generating one scattered ray per interaction.
The hardest part to debug is the refracted ray. I usually first just have all the light refract if
there is a refraction ray at all. For this project, I tried to put two glass balls in our scene, and
I got this (I have not told you how to do this right or wrong yet, but soon!):
Is that right? Glass balls look odd in real life. But no, it isn’t right. The world should be
flipped upside down and no weird black stuff. I just printed out the ray straight through the middle
of the image and it was clearly wrong. That often does the job.
The refraction is described by Snell’s law:
$$ n \cdot sin(theta) = n' \cdot sin(theta') $$
Where $n$ and $n'$ are the refractive indices (typically air = 1, glass = 1.3–1.7, diamond = 2.4)
and the geometry is:
One troublesome practical issue is that when the ray is in the material with the higher refractive
index, there is no real solution to Snell’s law and thus there is no refraction possible. Here all
the light is reflected, and because in practice that is usually inside solid objects, it is called
“total internal reflection”. This is why sometimes the water-air boundary acts as a perfect mirror
when you are submerged. The code for refraction is thus a bit more complicated than for reflection:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
bool refract(const vec3& v, const vec3& n, float ni_over_nt, vec3& refracted) {
vec3 uv = unit_vector(v);
float dt = dot(uv, n);
float discriminant = 1.0 - ni_over_nt*ni_over_nt*(1-dt*dt);
if (discriminant > 0) {
refracted = ni_over_nt*(uv - n*dt) - n*sqrt(discriminant);
return true;
}
else
return false;
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
And the dielectric material that always refracts when possible is:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
class dielectric : public material {
public:
dielectric(float ri) : ref_idx(ri) {}
virtual bool scatter(const ray& r_in, const hit_record& rec,
vec3& attenuation, ray& scattered) const
{
vec3 outward_normal;
vec3 reflected = reflect(r_in.direction(), rec.normal);
float ni_over_nt;
attenuation = vec3(1.0, 1.0, 0.0);
vec3 refracted;
if (dot(r_in.direction(), rec.normal) > 0) {
outward_normal = -rec.normal;
ni_over_nt = ref_idx;
}
else {
outward_normal = rec.normal;
ni_over_nt = 1.0 / ref_idx;
}
if (refract(r_in.direction(), outward_normal, ni_over_nt, refracted))
scattered = ray(rec.p, refracted);
else {
scattered = ray(rec.p, reflected);
return false;
}
return true;
}
float ref_idx;
};
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Attenuation is always 1 -- the glass surface absorbs nothing. The `attenuation = vec3(1.0, 1.0,
0.0)` above will also kill the blue channel which is the type of color bug that can be hard to find
-- it will give a color shift only. Try it the way it is and then change it to
`attenuation = vec3(1.0, 1.0, 1.0)` to see the difference.
If we try that out with these parameters:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
list[0] = new sphere(vec3(0,0,-1), 0.5, new lambertian(vec3(0.1, 0.2, 0.5)));
list[1] = new sphere(vec3(0,-100.5,-1), 100, new lambertian(vec3(0.8, 0.8, 0.0)));
list[2] = new sphere(vec3(1,0,-1), 0.5, new metal(vec3(0.8, 0.6, 0.2), 0.0));
list[3] = new sphere(vec3(-1,0,-1), 0.5, new dielectric(1.5));
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We get:
(The reader Becker has pointed out that when there is a reflection ray the function returns `false`
so there are no reflections. He is right, and that is why there are none in the image above. I
am leaving this in rather than correcting this because it is a very interesting example of a major
bug that still leaves a reasonably plausible image. These sleeper bugs are the hardest bugs to
find because we humans are not designed to find fault with what we see.)
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 simple and
surprisingly simple polynomial approximation by Christophe Schlick:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
float schlick(float cosine, float ref_idx) {
float r0 = (1-ref_idx) / (1+ref_idx);
r0 = r0*r0;
return r0 + (1-r0)*pow((1 - cosine),5);
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This yields our full glass material:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
class dielectric : public material {
public:
dielectric(float ri) : ref_idx(ri) {}
virtual bool scatter(
const ray& r_in, const hit_record& rec, vec3& attenuation,
ray& scattered) const
{
vec3 outward_normal;
vec3 reflected = reflect(r_in.direction(), rec.normal);
float ni_over_nt;
attenuation = vec3(1.0, 1.0, 1.0);
vec3 refracted;
float reflect_prob;
float cosine;
if (dot(r_in.direction(), rec.normal) > 0) {
outward_normal = -rec.normal;
ni_over_nt = ref_idx;
cosine = ref_idx * dot(r_in.direction(), rec.normal)
/ r_in.direction().length();
}
else {
outward_normal = rec.normal;
ni_over_nt = 1.0 / ref_idx;
cosine = -dot(r_in.direction(), rec.normal)
/ r_in.direction().length();
}
if (refract(r_in.direction(), outward_normal, ni_over_nt, refracted)) {
reflect_prob = schlick(cosine, ref_idx);
}
else {
reflect_prob = 1.0;
}
if (random_double() < reflect_prob) {
scattered = ray(rec.p, reflected);
}
else {
scattered = ray(rec.p, refracted);
}
return true;
}
float ref_idx;
};
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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, so it can be used as a bubble
to make a hollow glass sphere:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C++
list[0] = new sphere(vec3(0,0,-1), 0.5, new lambertian(vec3(0.1, 0.2, 0.5)));
list[1] = new sphere(vec3(0,-100.5,-1), 100, new lambertian(vec3(0.8, 0.8, 0.0)));
list[2] = new sphere(vec3(1,0,-1), 0.5, new metal(vec3(0.8, 0.6, 0.2)));
list[3] = new sphere(vec3(-1,0,-1), 0.5, new dielectric(1.5));
list[4] = new sphere(vec3(-1,0,-1), -0.45, new dielectric(1.5));
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This gives:
