conic

Conic, or conic section, is the curve formed by intersecting a plane with a double-napped cone. Depending on the angle and position of the cutting plane, the intersection is a circle, ellipse, parabola, or hyperbola. Degenerate cases occur when the plane passes through the cone’s apex, producing a point, a line, or a pair of intersecting lines.

Historically, the subject dates to Apollonius of Perga, who studied these curves as conic sections; the term conic derives from the Greek konos (cone).

Key geometric properties: Each conic can be defined as the locus of points with a fixed ratio (eccentricity e) of distances to a focus and a directrix. e=0 gives a circle, 0<e<1 an ellipse, e=1 a parabola, e>1 a hyperbola. Ellipses have two foci; parabolas have a single focus and a directrix; hyperbolas have two foci and two branches.

Analytic forms: In Cartesian coordinates, conics are second-degree equations Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 with discriminant B^2 - 4AC determining type. Circles: (x-h)^2+(y-k)^2=r^2. Ellipses: (x-h)^2/a^2+(y-k)^2/b^2=1. Parabolas: (x-h)^2=4p(y-k) or (y-k)^2=4p(x-h). Hyperbolas: (x-h)^2/a^2-(y-k)^2/b^2=1. The eccentricity can also be read from a and b, e=√(1−b^2/a^2) for ellipses (a≥b), and e=√(1+b^2/a^2) for hyperbolas.

Applications and notes: Conics arise in celestial mechanics as orbital paths of bodies, in optics and engineering, and in architecture and computer graphics. Degenerate cases and tangent properties are also considered in surveying and design.