do {

scattered_direction = vec3(x: -1.0 + 2.0*drand48(), y: -1.0 + 2.0*drand48(), z: -1.0 + 2.0*drand48())

} while dot(scattered_direction,scattered_direction) > 1.0

Then I stick a loop around that to test until

while dot(scattered_direction, hit_info.2) < 0.001

Here hit_info.2 is the surface normal. I could go in and produce correct diffuse rays, but that would involve coordinate systems and all that. Instead I wondered if I could use the trick shown on the right below:

do {

scattered_direction = vec3(x: -1.0 + 2.0*drand48(), y: -1.0 + 2.0*drand48(), z: -1.0 + 2.0*drand48())

} while dot(scattered_direction,scattered_direction) > 1.0

let sphere_tangent_point = -unit_vector(hit_info.2)

scattered_direction = scattered_direction - sphere_tangent_point

And this yields:

uniform rays |

diffuseish rays |

Looks like darker shadows which makes sense: rays go straight up. It would require some calculus to see if the rays are Lambertian, and this was an exercise to avoid work so I am not doing that. My money is on it being more oriented to the normal than true diffuse, but not bad.

## 5 comments:

I came up with the same technique some years back. My reasoning was that a cosine plotted in polar coordinates looks like a circle, which would be the equivalent of a cosine distribution in 2D. In 3D that would be it would look like a sphere (mumble, handwave, something about the cross section and rotational symmetry around the normal). Treat that sphere as a PDF, uniformly sample inside the sphere, and you're using rejection sampling to generate directions with a cosine distribution about the normal. I didn't try a more formal proof, but it looked good enough in the toy path tracer I was writing.

I like that logic. My only concern is the "beams" in a certain direction get fatter so it may grow faster than cosine theta. Let's put our programming hat on and figure out a way to test it empirically. Some numercal integration of a known integral maybe?

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I don't understand why you say the method on the right is probably closer to being Lambertian than the one on the left. Isn't a Lambertian BRDF simply one that "scatters incident illumination equally in all directions"? (Physically Based Rendering, 2nd ed, Pharr & Humphreys, p. 446.) So isn't the one on the left (the hemisphere) actually just purely Lambertian, ignoring any side effects of drand48? Is it maybe because the sampling occurs inside the volume of a sphere rather than on the surface of a sphere?

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