conicos

Conics, or conic sections, are the curves obtained by intersecting a plane with a double-napped cone. They include the circle, ellipse, parabola, and hyperbola. The study of conics traces to Apollonius of Perga in antiquity and was refined during the Renaissance, with the term conic sections popularized in analytic geometry.

Classification: The nondegenerate conics are four types: circle, ellipse, parabola, and hyperbola. A circle is a special ellipse with equal axes; a parabola is the locus of points equidistant from a focus and a directrix; an ellipse is the locus of points with a constant sum of distances to two foci; a hyperbola is the locus of points with a constant difference of distances to two foci. Degenerate cases include a point, a line, or two intersecting lines.

Algebraically, a conic is described by a second-degree equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. The discriminant B^2 − 4AC classifies the type: ellipse if negative, parabola if zero, hyperbola if positive. Standard forms include circle (x−h)^2+(y−k)^2=r^2; ellipse (x−h)^2/a^2+(y−k)^2/b^2=1; parabola y−k = p(x−h)^2; and hyperbola (x−h)^2/a^2 − (y−k)^2/b^2 = 1.

Focus-based definition: the locus of points for which the distance to a focus divided by the distance to a directrix equals a constant e, the eccentricity (e<1 for ellipse, e=1 for parabola, e>1 for hyperbola). The foci, directrices and eccentricity provide geometric insight and underlie properties such as the reflection behavior of conics in optics.

Conics are fundamental in analytic geometry and appear in astronomy, physics, engineering and computer graphics. They describe planetary orbits, lens shapes, and curves in architectural design, illustrating how a simple construction yields diverse and important curves.