# Geometric Factor

## Factor, Geometric

in photometry, a parameter that specifies the geometry of a beam of radiation.

The geometric factor G depends only on the sizes and mutual spacing of the diaphragms, which together separate out from all possible straight lines in space the set of directions that defines the ray or (if the region occupied by the light is of finite size) the beam of the radiation. The geometric factor is identical for all surfaces intersected by the straight lines in the given set (that is, it is invariant with respect to them) and is used as the measure of the set.

In the case of conjugate entrance and exit diaphragms *A _{s}* and

*A*of an optical system, for example, we have

_{d}*d*^{2}*G* = *dA _{s}* cos θ

_{s}

*d*Ω

_{s}=

*dA*cos θ

_{d}_{d}

*d*Ω

_{d}

where *d ^{2}G* is the second differential of the geometric factor,

*dA*and

_{s}*dA*are the areas of conjugate regions of the diaphragms or of the source and detector, θ

_{d}_{s}and θ

_{d}are the angles between the direction of the radiation and the normals to the emitting and illuminated surfaces, and

*d*Ω

_{s}and

*d*Ω

_{d}are the solid angles filled with radiation on the

*A*side and

_{s}*A*side.

_{d}The invariance of the geometric factor holds even for wide beams of light. The geometric factor is used for constructing systems of photometric quantities. For example, the luminance along a ray is *L* = *d*^{2}φ/*d*^{2}*G*, where φ is the luminous or radiant flux.

The concept of the measure of a set of rays was first introduced by the Soviet scientist A. A. Gershun in the 1930’s.

### REFERENCES

Gershun, A. A. “Mera mnozhestva luchei.”*Trudy Gosudarstvennogo opticheskogo in-ia*, 1941, vol. 14, issues 112–20.

Terrien, J., and F. Desvignes.

*La Photomé’lrie*. Paris, 1972.

A. A. VOL’KENSHTEIN