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In a Whitted ray tracer, each ray-object intersection spawns a transmitted ray (if the object was translucent), a reflected ray and a shadow ray. The shadow ray contributes the direct lighting component.

But what happens if the shadow ray intersects a transparent object? Is the direct lighting component ignored? How will diffuse objects submerged in water be lit if they don't get any direct light contributions from the shadow ray?

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You should first refer to Rendering Equation. It is the general equation to describe the physically transmitting of light in the view of computer science.

Whitted model is just an approximation of the surface integration of the Rendering Equation. It only calculates three light rays(shadow ray, reflected ray, and refracted ray). In a more sophisticated ray tracer, you should use Monte-Carlo ray tracing, where at every intersection point on the object, you will use thousands of rays sampling according to BRDF. Such backward algorithms actually work not good at caustic scene, which is the scene you say. You can use Photon Mapping plus Monte-Carlo Ray Tracing to get a better visualization.

If you just want to use Whitted model, you can multiply the lighting of shadow ray by a factor defined by the occluded transparent object.

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