Explore the cosine law of illumination: how the angle of incidence changes the illuminance a surface receives. Adjust intensity, distance and angle to see the geometry and calculation update live.
PHOTOMETRY SIMULATOR
Light ray and receiving surface
Live diagram
The angle θ is measured between the incoming light ray and the surface normal (perpendicular to the flat receiving surface). The source moves as θ and distance change; the glow and beam strength follow luminous intensity. Diagram distances are illustrative, while the calculation uses the entered values.
Adjust the experiment
Inputs
90° · Left0° · Normal90° · Right
m
0.25 m15 m
cd
1 cd1000 cd
Illuminance (E)
lux (lx)
Cosine factor (cos θ)
Illuminance relation
E = I cos θ / r²
The cosine law of illumination
For a point source, illuminance on a plane is proportional to the cosine of the angle between the incident ray and the surface normal. Including distance, the relation is E = I cos θ / r².
θ = 0°: cos θ = 1, so illuminance is maximum.
θ = 60°: cos θ = 0.5, so illuminance is half the normal-incidence value at the same distance.
θ = 90°: cos θ = 0, so ideal direct illuminance on the plane approaches zero.
Try these experiments
Keep distance and intensity fixed; change the angle from 0° to 90°.
Set θ = 0° and double the distance. Illuminance becomes one-quarter.
Keep the geometry fixed and double luminous intensity. Illuminance doubles.
Model assumptions: ideal point source, unobstructed direct light, a flat receiving surface, and no reflections or ambient light. Strictly, Lambert's emission law (intensity of an ideal diffuse surface ∝ cos θ) is a related but different statement; this page shows the illumination form of the cosine law.