I have been reading the book "real-time rendering 4th edition" and came across this Directional-hemispherical reflectance equation.

$$R(\mathbf{l}) = \int_{\mathbf{v}\in\Omega} f(\mathbf{l}, \mathbf{v})(\mathbf{n}\cdot\mathbf{v}) \,d\mathbf{v} $$

It can be used to measure to what degree a BRDF is energy conserving. According to my understanding what it does is that for a given light direction l we evaluate the BRDF for this l and every possible view direction in the upper hemisphere oriented along my surface normal n. In other words we capture the reflected light from every possible view sum that and see if the result we have is less than or equal to 1 (assuming a fixed radiance of 1). If it is then our BRDF is energy conserving as the outgoing energy must be less than or equal to the incoming energy (not counting glowing surfaces). What I don't get is the role of the NdotV in this equation I know that it represents the angle between the normal surface and the view direction but what I don't get is how that affects the amount of reflected light why having an angle of 90 degrees evaluates to 0 reflected light in this view direction.

  • $\begingroup$ Just as food for thought...when I see the dot product between to normalized vectors in lighting equations my mind jumps to cosine of the angle. In this case it is cosine of the angle between the normal vector and view vector... $\endgroup$
    – pmw1234
    Aug 12, 2022 at 10:39
  • $\begingroup$ Yes, the NdotV here represents cosine the angle between the view direction and normal direction, but i don't get the purpose of it in the integral, what does it represent? why when looking at the surface from 90-degree angle I get zero reflections. $\endgroup$ Aug 12, 2022 at 14:06

1 Answer 1


First, you are not exactly capturing "reflected light from view direction". What you mean is called importance, and it is transported exactly the same way in the scene as radiance (Helmholtz reciprocity), but in the opposite direction (i.e. from camera to light source).

Also note that for BRDF to be energy conserving, the equation must hold for every l.

Due to the reciprocity, the equation can be also rewritten with l and v interchanged, which can be more practical (see white furnace test).

The cosine of angle between a viewing direction and normal represents foreshortening, that is the reduction of area of the surface patch when projected into the viewing direction.


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