Coniche

Coniche are curves formed by intersecting a plane with a right circular double cone. The nondegenerate intersections produce circle, ellipse, parabola, or hyperbola. The specific curve is determined by the angle between the plane and the cone’s axis (or by the plane’s distance from the apex). A plane perpendicular to the axis cuts a circle; a plane tilted but not parallel to any generating line yields an ellipse; a plane parallel to a generating line yields a parabola; a plane steeper than the generating lines yields a hyperbola. If the plane passes through the apex, the intersection is degenerate, resulting in a point, a line, or a pair of lines.

In analytic geometry, conics satisfy a general second-degree equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, with discriminant B^2 − 4AC determining the type. Standard canonical forms include the circle x^2 + y^2 = r^2; ellipse x^2/a^2 + y^2/b^2 = 1; parabola y^2 = 4px (or x^2 = 4py); and hyperbola x^2/a^2 − y^2/b^2 = 1 (or y^2/b^2 − x^2/a^2 = 1). Conics admit focus–directrix definitions and eccentricity e, where e < 1 for ellipses, e = 1 for parabolas, and e > 1 for hyperbolas; the circle is a special ellipse with e = 0.

Historically, conic sections were investigated by Apollonius of Perga in antiquity, giving them their name from the Greek konikos, “cone-like.” The analytic treatment was later developed by Descartes. In astronomy, Kepler showed that planetary orbits are conic sections.

Applications include optics, celestial mechanics, architecture, CAD, and computer graphics.