cusp 
(kŭsp)
(kŭsp)n.
Idiom: 1. A point or pointed end.
2. Anatomy
a. A pointed or rounded projection on the chewing surface of a tooth.
b. A triangular fold or flap of a heart valve.
3. Mathematics A point at which a curve crosses itself and at which the two tangents to the curve coincide.
4. Architecture The point of intersection of two ornamental arcs or curves, such as the inner points of a trefoil.
5. Astronomy Either point of a crescent moon.
6. The boundary between two astrological signs or Zodiacal constellations.
on the cusp
1. On the threshold or verge of a development or action: an actor on the cusp of becoming a star.
2. At the dividing line or border of two conditions or categories: an artist on the cusp between Victorianism and modernism.
[Latin cuspis, point.]
cusp