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cusp
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cusp Scheitelpunkt {m}; Spitze {f} [listen]

point of osculation; tacnode; double cusp (of the second kind) (when two branches of a curve touch each other) Selbstberührungspunkt {m} (Knotenpunkt zweiter Art) [math.]

on the cusp of sth. an der Schwelle zu etw.
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