Consider a disk of radius s in the plane, centered at the origin, and the point P = (0, 0, -h) that's distance h below the disk. Assume that the disk is a Lambertian emitter, emitting radiance L in every direction from every point, and is the only surface in the scene.

(a) Write out an integral for the irradiance at P.

(b) Evaluate the integral. Switching to polar coordinates on the plane will help.

(c) Show that if h < s/10, then the irradiance at P is essentially the same as it would be if the disk covered the entire plane (i.e., if s were very large). Thus, for small values of h, the irradiance is nearly constant.

(d) Show that for h > 4s, the irradiance is within 1% of πL(s/h)²