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The Basic Lighting Laws:

The physical laws of lighting determine how much up-light is generated. The well known basic laws that apply to outdoor lighting are presented here for review.

Consider light coming from a fixture or any other light source from above:



Where:
H = Vertical Height (the mounting height) of the light-source above the ground.
D = Linear Distance from the source to the ground.
Θ = Angle between the vertical and the direction of D where the light strikes the ground.
x = Horizontal Distance between the point directly below the source to the point in question.

Inverse Square Law: Light reduction with increasing distance.
If the surface is normal to the direction of the incident light, then the illumination E at a point on a surface varies directly with the luminous intensity I of the source and inversely with the square of the distance D from the source: E = I/D2.

Lambert's Law or the Cosine Law: Light reduction with increasing angle of incidence.
If the surface at the point receiving the light is not normal to the source, the illumination will vary with the cosine of the angle of incidence. Combining this with the Inverse Square Law yields: E = I cosΘ/D2.

Cosine Cubed Rule: By substituting H/cosΘ for D we get: E = I cos3Θ/H2. Thus the illumination E at any point on the ground with angle of incidence Θ can be expressed simply as a function of the luminous intensity I of the source and its mounting height H.

Graphically: By plotting E versus Θ, we can see the drastic drop-off of the light into vertical angles above and beyond 75°.

For E = I/H2 cos3Θ, where I is the luminous intensity and H is the lamp height.

 Incidence
   angle       cos3Θ
      0°         1.00
     10°        0.955
     20°        0.830
     30°        0.650
     40°        0.450
     50°        0.266
     60°        0.125
     70°        0.040
     80°        0.005
     85°        0.001
     90°        0.00



From the graph above, you can see the reason why the refractor was created for cobra-heads back in the 1950s.





This graph compares the illumination drop-off for a good (ideal) full-cutoff (FCO) head (blue curve) with that of a classic cobra-head with refractor (red curve). Note the shaded area on the right. This is the portion of direct up-light from a fixture.



Next, reflected light versus direct up-light.



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