a point or pointed end.
Anatomy, Zoology, Botany. a point, projection, or elevation, as on the crown of a tooth.
Also called spinode. Geometry. a point where two branches of a curve meet, end, and are tangent.
Architecture. a decorative device, used especially in Gothic architecture to vary the outlines of intradoses or to form architectural foils, consisting of a pair of curves tangent to the real or imaginary line defining the area decorated and meeting at a point within the area.
Astronomy. a point of a crescent, especially of the moon.
the zodiacal degree that marks the beginning of a house or a sign.
Informal. a person born on the first day of a sign.
a point that marks the beginning of a change: on the cusp of a new era.

1575–85; < Latin cuspis a point

cuspal, adjective Unabridged
Based on the Random House Dictionary, © Random House, Inc. 2014.
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World English Dictionary
cusp (kʌsp)
1.  any of the small elevations on the grinding or chewing surface of a tooth
2.  any of the triangular flaps of a heart valve
3.  a point or pointed end
4.  geometry Also called: spinode a point at which two arcs of a curve intersect and at which the two tangents are coincident
5.  architect a carving at the meeting place of two arcs
6.  astronomy either of the points of a crescent moon or of a satellite or inferior planet in a similar phase
7.  astrology any division between houses or signs of the zodiac
[C16: from Latin cuspis point, pointed end]

Collins English Dictionary - Complete & Unabridged 10th Edition
2009 © William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins
Publishers 1998, 2000, 2003, 2005, 2006, 2007, 2009
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Word Origin & History

1580s, from L. cuspis "point, spear, pointed end." Astrological use is earliest.
Online Etymology Dictionary, © 2010 Douglas Harper
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American Heritage
Medical Dictionary

cusp (kŭsp)

  1. A pointed or rounded projection on the chewing surface of a tooth.

  2. A triangular fold or flap of a heart valve.

The American Heritage® Stedman's Medical Dictionary
Copyright © 2002, 2001, 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company.
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Example sentences
My life's dream of becoming a professor was on the cusp of coming true.
We are on the cusp of a change in our understanding of the nature of all
  things, and human beings in particular.
So already the reader is at several veiled removes even as the speaker
  intimates that she's on the cusp of making a revelation.
More and more too are withdrawing to care for elderly parents at a time when
  they are on the cusp of the higher echelons.
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