Algebraic geometry
The main objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of sets of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves and quartic curves like lemniscates, and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates match a given polynomial equation. Simple questions involve the study of points of special interest like the singular points, the inflection points and the points at infinity. More difficult questions involve the topology of the curve and relations between the curves given by different equations.