'curve-based' is a compound formed from the noun/adjective 'curve' and the past-participle/adjective 'based'. 'curve' originates from Late Middle English, from Old French (curue/curv-) and ultimately from Latin 'curvus' meaning 'bent'. 'based' comes from the verb 'base', which in the sense 'to found or establish' has roots in Old French and Late Latin (from Greek 'basis' in related senses) meaning 'foundation'.

'curve' developed from Latin 'curvus' > Old French 'curue/curv-' > Middle English 'curve', eventually stabilizing as modern English 'curve'. 'base' (verb/adjective) evolved via Old French (e.g. 'baser') and Late Latin/Greek influence (e.g. 'basis') into Middle English 'base' and the past participle form 'based'. The compound 'curve-based' is a modern English formation combining these elements to describe something founded on curves.

Originally 'curve' meant 'bent' and 'base' (from 'basis') referred to a foundation; over time the combined compound came to mean 'founded on or using curves' in technical and descriptive contexts.