Lambert's cosine law

src: breeds-of-horses.com

In optics, Lambert cosine law says that the intensity of jets or the intensity of light observed from the ideal diffusion surface or the ideal diffuse radiator is directly proportional to the cosine of the angle ? between the direction of the incident light and the normal surface. This law is also known as cosine emission law or Lambert emissions law . Named after Johann Heinrich Lambert, from Photometria , published in 1760.

The surface that obeys Lambert's law is said to be Lambertian , and shows Lambertian reflection. Such surfaces have the same jets when viewed from different angles. This means, for example, that for the human eye it has the same clear brightness (or luminance). It has the same emission because, although the power emitted from a particular area element is reduced by the cosine of the emission angle, the apparent size of the observed area (= "projected source area") as seen by the viewer decreases by the corresponding number. Or equivalent, the remote sensor with a constant solid viewing angle (or "aperture") will see a larger source area with decreased emission angle, but observe the less radiated power per unit source area: these two effects offset each other and hence rays which are observed independently from the point of emission. Therefore, the emission (power per unit of solid angle per unit of projected source area) is the same.

Video Lambert's cosine law

Lambertian pencar dan radiator

When the area element radiates from being illuminated by an external source, the radiation (energy or photon/time/area) radiation on the area element will be proportional to the angular cosine between the illuminated and normal sources. A scattering of Lambertian would then spread this light according to the same cosine law as Lambertian's producer. This means that although the surface emission depends on the angle from the normal source to the illuminating source, it will not depend on the angle from normal to the observer. For example, if the moon is Lambertian scattering, one would expect to see the scattered brightness sufficiently decrease toward the terminator because of the increased angle at which sunlight hits the surface. The fact that it does not diminish illustrates that the moon is not a Lambertian hunter, and actually tends to spread more light into a slanting corner than a Lambertian hunter.

Lambertian radiator emission does not depend on the amount of incident radiation, but from the radiation coming from the radiating body itself. For example, if the sun is a Lambertian radiator, one would expect to see constant brightness throughout the sun disc. The fact that the sun indicates the darkening of branches in the visible region illustrates that it was not Lambertian radiators. The black body is an example of Lambertian radiator.

Maps Lambert's cosine law

Details of the same brightness effect

The situation for Lambertian surfaces (emitting or scattering) is illustrated in Figures 1 and 2. For conceptual clarity we will think in terms of photons rather than radiant energy or energy. The wedges in each circle represent the same angle d? , and for the Lambertian surface, the number of photons per second transmitted to each slice is proportional to the area of ​​the slice.

The length of each slice is the product of the circle diameter and cos (? ). The maximum rate of photon emissions per unit of solid angle is normal, and decreases to zero for ? = 90Ã, Â °. In mathematical terms, the light throughout normal is I photons/(sÃ, Â · cm ² Ã, Â · sr) and the number of photons per second transmitted into vertical slices is I d? dA . The number of photons per second transmitted into the slices at the angle ? is I cos (? ) Ã, d? dA .

Figure 2 shows what the observer sees. An observer just above the area element will see the view through the area dA ₀ and the area element dA will insert (solid) > d? ₀ . We can assume without losing the announcement that the aperture occurs to form a solid angle d? when "viewed" of the transmitter element element. This normal observer will then record I d? dA photons per second and so will measure the emission of light

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂI_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â0Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\frac{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂIÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂdÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\mathrm{?}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂdÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂAÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂdÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â{\mathrm{?}}_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â0Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂdÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂA_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â0Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; ÂAnnotation\; encoding\; =\; "application/x-tex">\; \{\backslash \; displaystyle\; I\_\; \{0\}\; =\; \{\backslash \; frac\; \{I\; \backslash ,\; d\; \backslash \; Omega\; \backslash ,\; dA\}\; \{d\; \backslash \; Omega\; \_\; \{0\}\; \backslash ,\; dA\_\; \{\; 0\}\}\}\}\; Â\; Â$ foton/(sÃ, Â · cm ² Ã, Â · sr).

An observer at the angle of ? to normal will see the scenes through the same gap of the area dA ₀ and the area element dA i> d? ₀ cos (? ). This observer will record I cos (? ) d? dA photons per second, and so will measure the light

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂI_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â0Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\frac{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂIÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\cos Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â()\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â?}{}}$                             d               ?                             d              A                                       d                               ?                                   0                                                           cos                             ()               ?                             d                               A                                   0                                                                     =                                             I                             d               ?                             d              A                                       d                               ?                                   0                                                           d                               A                                   0                                                                       {\ displaystyle 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}}}} foton/(sÃ,  · cm ² Ã,  · sr),

the same as the normal observer.

src: i.ytimg.com

Connecting the peak luminous intensity and luminous flux

dan sebagainya

${\ displaystyle F_ {tot} = \ pi \, \ mathrm {sr} \ cdot I_ {max}}  ÂÂ$

di mana ${\ displaystyle \ sin (\ theta)}  ÂÂ$ adalah determinan matriks Jacobian untuk unit sphere, dan menyadari bahwa ${\ displaystyle I_ {max}}  ÂÂ$ adalah luminous flux per steradian. Demikian pula, intensitas puncak akan ${\ displaystyle 1/(\ pi \, \ mathrm {sr})}  ÂÂ$ dari total luminous flux yang terpancar. Untuk permukaan Lambertian, faktor yang sama ${\ displaystyle \ pi \, \ mathrm {sr}}  ÂÂ$ menghubungkan luminansi dengan pancaran cahaya, intensitas pancaran ke fluks radiasi, dan pancaran sinar ke pancaran pancaran. Radians dan steradians, tentu saja, tidak berdimensi dan jadi "rad" dan "sr" dimasukkan hanya untuk kejelasan.

Example: The surface with luminance says 100 cd/m ² (= 100 nits, regular PC monitor) will, if it is a perfect Lambert emitter, has a light transmit power of 314 lm/m ² . If the area is 0.1 m ² (~ 19 "monitor) then the total emitted light, or luminous flux, will be 31.4 lm.

src: lightinganalysts.com

Usage

Lambert's cosine law in its reversed form (Lambertian Reflection) implies that the apparent brightness of the Lambertian surface is proportional to the cosine of the angle between the normal surface and the direction of incident light.

This phenomenon can be used when making prints, with the effect of creating bright and dark lines on structures or objects without having to change materials or apply pigments. The contrast of dark and bright areas defines the object. Moldings are strips of material with various cross sections used to cover transitions between surfaces or for decoration.

src: i.ytimg.com

See also

• Casting
• Reflective Power
• Passive solar building design
• Sun path

src: i.ytimg.com

References

Source of the article : Wikipedia