omission of circle vowels

In working through the 5,000 most used forms (Anniversary), I came across things I don’t understand. Could someone explain to me why circle vowels are sometimes ommitted even when they occur in accented syllables?

Examples:
pen – no “e”
pig – no “i”
pity – no “i”
bid – no “i”
beauty – no “eau”

but:
pan, pin, peg, pick, pit, and bit all have the vowel

On a related note, why does “happiness” have the joining it does? I thought circle vowels were supposed to go outside of angles (p and n).

Thanks a bunch!

Brian