'curvilinear-based' is an English compound formed from 'curvilinear' and 'based'. 'Curvilinear' ultimately derives from Latin elements: 'curvus' meaning 'bent' and 'linea' meaning 'line'. 'Based' comes from the noun 'base' (via Old French 'base') from Latin/Greek 'basis' meaning 'foundation'.

'Curvilinear' developed in modern English from the combining form 'curvi-' (from Latin 'curvus') + 'linear' (from Latin 'linea') to describe curved-line geometry; 'base' entered English via Old French 'base' from Late Latin/Greek 'basis', and the adjectival compound 'X-based' arose in modern English to mean 'having X as its basis'. The compound 'curvilinear-based' is a contemporary formation combining these elements.

Individually, 'curvilinear' originally described 'having curved lines' and 'base' meant 'foundation'; combined in modern usage they specifically convey 'founded on or using curved/curvilinear forms'.