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54 MATHEMATICAL TERMS CHAPTER 4 4.2 RADIANCE To account for this effect of the apparent area decreasing with view direction, the key quantity radiance in a particular direction is defined as the radiant flux per unit solid angle and unit area projected in the direction . The radiance L is defined as: L( ) = d 2 ( )/cos dA d (4.8) Understanding the cos in the denominator of the definition of radiance is one of two difficult concepts in the mathematical description of how light interacts with materials. One way to think of its convenience, is to think of a small patch of diffuse material, such as the right ear of the dog shown in Figure 3.11a. Looking straight at the ear in the right image, the ear appears to have a certain brightness, and a certain amount of light energy from it reaches our eye. Looking at the ear at a more glancing angle in the left image, the patch is just as bright, but overall less light from it reaches our eye, because the area the light is coming from is smaller from our new point of view. The quantity that affects our perception of brightness is the radiance. The radiance of the patch for a diffuse material is the same for all directions even though the energy per unit time depends on the orientation of the view relative to the surface. As discussed in Chapter 2, we form an image by computing the light that arrives at a visible object through each pixel. We can make the statement "computing the light" more specific now, and say that we want to compute the radiance that would arrive from the visible object. The radiance has been defined so that if we compute the radiance at an object point in a particular direction, in clear air the radiance along a ray in that direction will be constant. We qualify this with "in clear air" since volumes of smoke or dust might absorb some light along the way, and therefore the radiance would change. The other variable that we want to account for in addition to time, position, and direction, is wavelength. Implicitly, since we are concerned with visible light in the span of 380 to 780 nm, all of the quantities we have discussed so far are for energy in that band. To express flux, irradiance, radiant exitance, intensity, or radiance as a function of wavelength, we consider the quantity at a value of within a small band of wavelengths between and + d . By associating a d with each value, we can integrate spectral values over the whole spectrum. We express the spectral quantities such as the spectral radiance as: L( , x, y, ) = d 3 ( , x, y, )/cos dA d d (4.9) To simplify this notation, we denote the two-dimensional spatial coordinate x, y with the boldface x. In many cases, we will want to indicate whether light is incident on or leaving the surface. Following Philip Dutre's online Global Illumination Compendium