Nice word cusp. Can't think why it isn't used more. It's not as though it doesn't have many uses.
A cusp is a pointed end where two points meet. This could be part of a tooth (pre molars are bicuspid as they have two pointy bits) or an architectural or biological feature. Eg the top of a gothic arch, a leap point or part of the valves in our hearts.
If a mathmatical curve reverses abruptly that change is marked by a cusp and cusp is also the name given to the pointed ends of a new moon.
Cusp can also denote a transition of some kind. Those who's birth sign is on or near the end of a period are said to be on the cusp. Eg if your birthday is on 20th March you're on the cusp of Pisces and Aries (what that means I haven't a clue!) Teenagers are sometimes considered on the cusp of adulthood (as are middle-aged writers!)