coniques

Conics, or conic sections, are the curves obtained by intersecting a right circular double cone with a plane. Depending on the angle and position of the plane, the intersection is a circle, an ellipse, a parabola, or a hyperbola. The circle is a special case of an ellipse when the plane is perpendicular to the cone's axis.

Equivalently, a conic can be defined as the locus of points whose distance to a fixed focus and to a fixed directrix are in a constant ratio, the eccentricity e. If e<1 the locus is an ellipse, if e=1 a parabola, if e>1 a hyperbola. A circle corresponds to e=0 or to an ellipse with equal axes.

In Cartesian form the standard curves are: circle (x-h)^2+(y-k)^2=r^2; ellipse (x-h)^2/a^2+(y-k)^2/b^2=1 with a≠b; parabola y-k = p(x-h)^2 (vertical) or x-h = p(y-k)^2 (horizontal); hyperbola (x-h)^2/a^2-(y-k)^2/b^2=1 or (y-k)^2/a^2-(x-h)^2/b^2=1.

Conics have symmetry axes, a center for ellipses and hyperbolas, and, in the case of a hyperbola, asymptotes. They arise in classical astronomy as the orbits of planets and comets (ellipses and hyperbolas as solutions to Kepler's laws), as well as in optics and engineering. The term conic is derived from Greek konkikos, and the study was central to Apollonius of Perga in ancient geometry.