.
The classic cube doubling, angle trisection and squaring the circle problems.
Trying to solve the classical problems of Antiquity, such as the trisection problem, Greeks discovered different mathematical curves. There was no systematic theory of higherdegree curves in Greek mathematics but Greeks studied many interesting special cases:

The Cissoid of Diocles ( c. 100 BC )
Diocles showed that the Cissoid a cubic curve defined by y^{2 }(1+x) = (1x)^{3 }could be used to duplicate the cube

The Spiric Sections of Perseus ( c. 150 BC )

The Epicycles of Ptolemy ( c. 140 AD ) used to describe the retrograde motion of Planets in his Earth centric model.

Ellipse, Parabola, Hyperbola (Conic Sections) discovered probably by Menaechmus and theory studied in details by Apollonius.
Xah Lee has a very interesting Website that includes also information of Curves among which some have been discovered by ancient Greek mathematicians. I provide the links to the corresponding plane curves. Later I will provide additional information. Also other links are provided.
CURVE 
LINKS 
Archimedean Spiral 

Conchoid of Nicomedes 

Cissoid of Diocles 

Spiric of Perseus 

General Conic Sections 

Ellipse 

Hyperbola 

Parabola 

Quadratix of Hippias 
A Crowning Achievement: The Quadratrix of Hippias Construction from a cylindrical spiral, as the intersection of a plektoid and a plane Construction from an Archimedes spiral, as the intersection of a plektoid and a plane 
Hippopede of Proclus (and Eudoxus) 

Philon's Line 
Derivation of the Formula for the Area of a Circle and the Pythagorean Theorem
LINKS
Vignettes of Ancient Mathematics by Henry Mendell, Cal. State U., L.A.
Ancient Greece 
Medieval Greece / Byzantine Empire 
Modern Greece 

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