Lambert's cosine law 41306 219959346 2008-06-17T17:14:22Z Tomruen 63601 [[WP:UNDO|Undid]] revision 219951997 by [[Special:Contributions/206.181.226.34|206.181.226.34]] ([[User talk:206.181.226.34|talk]]) {{Seealso|Lambertian reflectance}} '''Lambert's cosine law''' in [[optics]] says that the [[radiant intensity]] observed from a "'''Lambertian'''" surface is [[directly proportional]] to the [[cosine]] of the angle &theta; between the observer's line of sight and the [[surface normal]]. The law is also known as the '''cosine emission law''' or '''Lambert's emission law'''. It is named after [[Johann Heinrich Lambert]], from his ''Photometria'', published in 1760. An important consequence of Lambert's cosine law is that when such a surface is viewed from any angle, it has the same apparent [[radiance]]. This means, for example, that to the human eye it has the same apparent brightness (or [[luminance]]). It has the same radiance because, although the emitted power from a given area element is reduced by the cosine of the emission angle, the size of the observed area is increased by a corresponding amount, so that while the area element appears the same in reality it has increased by the cosine of the angle and therefore its radiance is the same. For example, in the [[visible spectrum]], the [[Sun]] is almost a perfect Lambertian radiator, and as a result the brightness of the Sun is almost the same everywhere on an image of the solar disk even though we see it full-on only in the center of the disk. Also, a [[black body]] is a perfect Lambertian radiator. == Lambertian scatterers == When an area element is radiating as a result of being illuminated by an external source, the [[irradiance]] (energy or photons/time/area) landing on that area element will be proportional to the cosine of the angle between the illuminating source and the normal. A Lambertian scatterer will then scatter this light according to the same cosine law as a Lambertian emitter. This means that although the radiance of the surface depends on the angle from the normal to the illuminating source, it will not depend on the angle from the normal to the observer. For example, if the moon were a Lambertian scatterer, one would expect to see its scattered brightness appreciably diminish towards the [[terminator (solar)|terminator]] due to the increased angle at which sunlight hit the surface. The fact that it does not diminish illustrates that the moon is not a Lambertian scatterer, and in fact tends to scatter more light into the [[oblique angle]]s than would a Lambertian scatterer. ==Details of equal brightness effect== <div style="float: right"> {| |- | [[Image:LambertCosineLaw1.png|frame|right|Figure 1: Emission rate (photons/s) in a normal and off-normal direction. The number of photons/sec directed into any wedge is proportional to the area of the wedge.]] |- | [[Image:LambertCosineLaw2.png|frame|right|Figure 2: Observed intensity (photons/(s·cm<sup>2</sup>·sr)) for a normal and off-normal observer; ''dA''<sub>0</sub> is the area of the observing aperture and ''d&Omega;'' is the solid angle subtended by the aperture from the viewpoint of the emitting area element.]] |} </div> The situation for a Lambertian surface (emitting or scattering) is illustrated in Figures 1 and 2. For conceptual clarity we will think in terms of [[photon]]s rather than [[energy]] or [[luminous energy]]. The wedges in the [[circle]] each represent an equal angle ''d &Omega;'' and, for a Lambertian surface, the number of photons per second emitted into each wedge is proportional to the area of the wedge. It can be seen that the length of each wedge is the product of the [[diameter]] of the circle and cos(''&theta;''). It can also be seen that the maximum rate of photon emission per unit [[solid angle]] is along the normal and diminishes to zero for ''&theta;'' = 90°. In mathematical terms, the [[radiance]] along the normal is ''I''&nbsp;photons/(s·cm<sup>2</sup>·sr) and the number of photons per second emitted into the vertical wedge is ''I'' ''d&Omega;'' ''dA''. The number of photons per second emitted into the wedge at angle ''&theta;'' is ''I''&nbsp;cos(''&theta;'')&nbsp;''d&Omega;''&nbsp;''dA''. Figure 2 represents what an observer sees. The observer directly above the area element will be seeing the scene through an aperture of area ''dA''<sub>0</sub> and the area element ''dA'' will subtend a (solid) angle of ''d&Omega;''<sub>0</sub>. We can assume without loss of generality that the aperture happens to subtend solid angle ''d&Omega;'' when "viewed" from the emitting area element. This normal observer will then be recording ''I''&nbsp;''d&Omega;''&nbsp;''dA'' photons per second and so will be measuring a radiance of :<math> I_0=\frac{I\, d\Omega\, dA}{d\Omega_0\, dA_0} </math> photons/(s·cm<sup>2</sup>·sr). The observer at angle ''&theta;'' to the normal will be seeing the scene through the same aperture of area ''dA''<sub>0</sub> and the area element ''dA'' will subtend a (solid) angle of ''d&Omega;''<sub>0</sub>&nbsp;cos(''&theta;''). This observer will be recording ''I''&nbsp;cos(''&theta;'')&nbsp;''d&Omega;''&nbsp;''dA'' photons per second, and so will be measuring a radiance of :<math> I_0=\frac{I \cos(\theta)\, d\Omega\, dA}{d\Omega_0\, \cos(\theta)\, dA_0} =\frac{I\, d\Omega\, dA}{d\Omega_0\, dA_0} </math> photons/(s·cm<sup>2</sup>·sr), which is the same as the normal observer. ==See Also== * [[Transmittance]] * [[Reflectivity]] * [[Passive solar building design]] * [[Sun path]] [[de:Lambertsches Gesetz]] [[it:Legge di Lambert]] [[Category:Radiometry]] [[Category:Photometry]] [[Category:3D computer graphics]]