Reflection From a Sphere

Assume a distant source uniformly illuminates a mirror sphere with a radius of 1 unit. What is the brightness per steradian of the reflection at angle \theta from the light source?

If we map angular space around the sphere in spherical coordinates, with θ as the angle of the reflection from the light source axis, and φ being the angle around the axis, then the light leaving at angle \theta is bounced off the sphere at angle \theta/2 The steradian area of the an angle element d\theta ~ d\phi is \sin \theta ~ d\theta ~ d\phi . This is associated with an intercepted light area on the sphere of 0.5 ~ sin( \theta ) ~ \sin( \theta / 2 ) ~ \cos( \theta / 2 ) ~ d\theta ~ d\phi or { 1 \over 4 } ( 1 - cos( \theta ) ) ~ sin( \theta ) ~ d\theta ~ d\phi

The light per steradian is thus

\left( { 1 \over 4 } ( 1 - cos( \theta ) ) ~ sin( \theta ) ~ d\theta ~ d\phi \right) / \left( sin( \theta ) ~ d\theta ~ d\phi ~\right)

or

{ 1 \over 4 } ( 1 - \cos( \theta ) )