Umbra

A total solar eclipse, or a complete blocking out of the Sun's light, can only be seen by those who live in the area covered by the umbra. People who live in the area of the Earth covered by the penumbra will see a partial eclipse.

In order to set up such an algorithm, we need the notion of multiplicative inverse of an umbra. Two umbrae are said to be multiplicative inverse to each other when αγ ≡ u.