We consider the rendering of diffuse objects under distant illumination,
as specified by an environment map.
Using an analytic expression for the irradiance in terms of
spherical harmonic coefficients of the lighting,
we show that one needs to compute and use only 9 coefficients,
corresponding to the lowest-frequency modes of the illumination,
in order to achieve average errors of only 1%.
In other words, the irradiance is insensitive to high frequencies in the lighting,
and is well approximated using only 9 parameters.
In fact, we show that the irradiance can be procedurally represented simply
as a quadratic polynomial in the cartesian components of the surface normal,
and give explicit formulae. These observations lead to
a simple and efficient procedural rendering algorithm amenable to hardware implementation,
a prefiltering method up to three orders of magnitude faster than previous techniques,
and new representations for lighting design and image-based rendering.