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curve has 20 distinct synonyms spread across 9 of its 10 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. Its one recorded opposite is straight line.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
the trace of a point whose direction of motion changes
a line on a graph representing data
a pitch of a baseball that is thrown with spin so that its path curves as it approaches the batter
the property possessed by the curving of a line or surface
curved segment (of a road or river or railroad track etc.)
Synonyms, antonyms and the broader and narrower relations for curve come from Princeton University’s WordNet. Full sources and licences.