conics

Conics are curves obtained by intersecting a plane with a right circular cone (a double-napped cone). They can also be defined as the locus of points such that the ratio of the distance to a fixed point (focus) and to a fixed line (directrix) is a constant called eccentricity. The resulting curves are the circle, ellipse, parabola, and hyperbola. A circle is a special ellipse with equal radii and a plane perpendicular to the cone's axis. An ellipse arises when the cutting plane intersects one nappe but is not parallel to any generator. A parabola occurs when the plane is parallel to a generator. A hyperbola results when the plane cuts through both nappes or is steep enough to intersect both.

Algebraically, a conic is described by a quadratic equation in two variables: Ax^2+Bxy+Cy^2+Dx+Ey+F=0 with discriminant B^2-4AC. The type is determined by this discriminant: negative for an ellipse (including circles with A=C and B=0), zero for a parabola, and positive for a hyperbola. With centered forms: circle (x-h)^2+(y-k)^2=r^2; ellipse (x-h)^2/a^2+(y-k)^2/b^2=1; parabola (x-h)^2=4p(y-k) or (y-k)^2=4p(x-h); hyperbola (x-h)^2/a^2-(y-k)^2/b^2=1 or swapped.

Degenerate conics include a single point, a line, or a pair of lines when the plane passes through the apex or has a special orientation.

Conics have a rich history in geometry, dating to Apollonius and earlier, and were central to Kepler’s description of planetary orbits. They remain fundamental in optics, engineering, and computer graphics due to their reflective and projective properties.