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### #ActualZBethel

Posted 27 May 2012 - 10:56 PM

I'm trying to wrap my head around the mathematics behind the BRDF. I think I understand Radiance and Irradiance: irradiance is the flux through a projected area. In the case of the BRDF, irradiance means the flow of energy through the projected area that represents the point on the surface where we are evaluating the lighting equation. Radiance is the flow of energy through the projected area with respect to a solid angle.

The BRDF is defined as fr(wi, wo) = dL(wi) / dE(wo), where wi and wo are the incident and exitant solid angles.

According to wikipedia, this equation expands to fr(wi, wo) = dL(wi) / L(wi) * cos(theta) * dw. From what I can tell, this is the differential outgoing radiance divided by the incoming radiance along the incident solid angle, multiplied by the cosine of the azimuth angle (to account for the projected area term), multiplied by the differential solid angle.

My question is, if L is the energy evaluated along the incident differential solid angle, what does the differential of Radiance mean? To clarify, why do we need dL when L is effectively giving us the light energy along a single ray of light? Also, in the denominator, we have the radiance term, L, multiplied by the cosine of the angle (which I understand to be projecting the incident radiance onto the surface). Why is that L instead of dL?

Thanks.

### #1ZBethel

Posted 27 May 2012 - 10:01 PM

I'm trying to wrap my head around the mathematics behind the BRDF. I think I understand Radiance and Irradiance: irradiance is the flux through a projected area. In the case of the BRDF, irradiance means the flow of energy through the projected area that represents the point on the surface where we are evaluating the lighting equation. Radiance is the flow of energy through the projected area with respect to a solid angle.

The BRDF is defined as fr(wi, wo) = dL(wi) / dE(wo), where wi and wo are the incident and exitant solid angles.

According to wikipedia, this equation expands to fr(wi, wo) = dL(wi) / L(wi) * cos(theta) * dw. From what I can tell, this is the differential outgoing radiance divided by the incoming radiance along the incident solid angle, multiplied by the cosine of the azimuth angle (to account for the projected area term), multiplied by the differential solid angle.

My question is, if L is the energy evaluated along the incident differential solid angle, what does the differential of Radiance mean? Also, in the denominator, we have the radiance term, L, multiplied by the cosine of the angle (which I understand to be projecting the incident radiance onto the surface). Why is that L instead of dL?

Thanks.

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