How to deal with refraction when the rays start inside of a nested object

Question

I am building a simple raytracer for educational purposes and want to add refraction to objects. Using Snells Law, I am able to create a new ray recursively at the intersection points. The raytracer supports currently only spheres and I use a scene where I have multiple spheres nested inside of each other with different refraction indices.

If I start a ray from outside of the spheres, everything seems simple. You start with the refraction index of the scene, and as soon as you hit the first sphere, refract the ray using the refraction index before and the refraction index of the material of the sphere until you hit the next sphere and so on. Using the normals of the intersection I can determine whether I enter or leave the sphere.

However, I don't understand how I should handle sphere leaves and what to do if the ray doesn't start in the outer part of the scene.

• Can I just take a stack of the refraction indices and go one layer up as soon as I leave a sphere?
• How can I determine with what refraction index I have to start if I start inside of the spheres?

Example

You have three spheres, with refraction indices 0.9, 1.1 and 0.8 from outer to inner. Air index is 1.0

• Your camera is outside of the sphere and points at the center of the sphere:

  • start index is 1.0, you first hit the outer sphere with index 0.9 and refract from 1.0 to 0.9 and save that your ray is now in 0.9 material
  • you hit the middle sphere and notice the material constant of 1.1, since you have saved the 0.9, you know that you have to refract from 0.9 to 1.1 and save the 1.1 in addition to the 0.9
  • you hit the inner sphere and refract from 1.1 to 0.8 and you have save until now 0.9, 1.1 and 0.8
  • you hit the inner sphere again (this time you exit it, so you check your saved values and know that you have to switch back to 1.1)
  • ... until you are outside

• Problem now, when the camera is inside the sphere. You won't know to what refraction index you have to switch.

Answer 1

Posting this from a physics point of view and not a raytracing implementation point of view :P.

Snell's law states that the ratio of the sines of the incident angle and the refractive angle are equal to the inverse of the ratio of the refractive index of the two mediums on either side of the boundary.

Thus, when you have a ray approaching a new material and you want to know the angle in the new material, you need to know the angle that the ray is hitting the new material, the index of refraction of the new material, and the index of refraction of the material the ray is currently in.

As you say refraction works fine moving into the sphere, you must know the index of refraction of each sphere and your scene already.

I'd say creating a stack of refractive indices would be a good way to deal with going into a bunch of nested materials, as you're going to have to touch all the refractive indexes that you push onto the stack again as you move out of the nested set of spheres.

As to determining what refractive index you have to start with as you leave the spheres, you're always saying sin(theta1)/sin(theta2) = [refractive index of 2]/[refractive index of 1]. Thus, you need the refractive index of the material that you're currently in and the index of the material that you're going to be moving towards.

Apologies if I misunderstood your question, but I hope that helps!