In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.
Hyperbolas as declination lines on a sundial
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.
This diagram clarifies the different angles of the cutting planes that result in the different properties of the three types of conic section.
Types of conic sections: 1: Circle 2: Ellipse 3: Parabola 4: Hyperbola
Diagram from Apollonius' Conics, in a 9th-century Arabic translation
Table of conics, Cyclopaedia, 1728
