Cusp
Etymology
From Latin cuspis, "a point, pointed end, the tip of a spear." It entered English as a technical term for a pointed end — the points of a crescent moon, the peak between two curves, the pointed part of a tooth. Astrology used it for the dividing point between two zodiac signs, and from there "on the cusp" came to mean poised at the edge of a change.
noun · /KUSP/
A point of transition between two states, or the edge just before a change.
What it names
The word for the brink of something about to happen — on the cusp of adulthood, of a breakthrough, of collapse — the moment poised between one condition and the next.
Example
The company was on the cusp of profitability when the funding suddenly dried up.
Related idea
Watershed is a dividing point recognized in hindsight, after the change has clearly taken effect; cusp is felt in the present, the tense edge just before the change arrives.
noun · /KUSP/
A pointed end or projection — a point of a crescent moon, a peak on the crown of a tooth, or where two arcs meet.
What it names
the literal sharp point — as on a molar, a leaf, or a Gothic arch
Example
The dentist noted wear on one cusp of the molar.
Related idea
The “point of transition” sense grows from the geometric one: a cusp is where a curve reverses, a knife-edge tip — the same sharpness behind standing on the cusp of change, at a threshold.