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Solid Angle and Projected Solid Angle

Excerpt from Field Guide to Illumination

The relationship between solid angle and projected solid angle can be confusing. Projected solid angle has meaning primarily for a small Lambertian source, which has intensity that varies as the cosine of the angle with the surface normal. The projected solid angle, Ω, is the solid angle, ω, weighted by the cosine of the angle with the surface normal.

When the solid angle is large enough so that the angle with the surface normal is not the same over the entire solid angle, the total projected solid angle must be computed by integrating the incremental projected solid angles. 

For some special cases, the integration results in simple expressions, such as for a large circular cone that is normal to a surface and subtends a half angle, θ.

A hemisphere has 2π steradians (solid angle) but π projected steradians (projected solid angle).

A. V. Arecchi, T. Messadi, and R. J. Koshel, Field Guide to Illumination, SPIE Press, Bellingham, WA (2007).

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