Conisch

Conisch is the adjective meaning conical or conic. In geometry it refers to shapes related to a cone or to the curves obtained by intersection of a plane with a double-napped cone. The term is used in mathematics, design, and related fields to describe both physical objects shaped like a cone and the mathematical curves known as conic sections.

Conic sections arise when a plane cuts through a cone. Depending on the angle and position of the cut, the intersection can be a circle, an ellipse, a parabola, or a hyperbola. The circle is a special case of the ellipse obtained when the cutting plane is perpendicular to the cone’s axis. Degenerate cases include a single point or a pair of intersecting lines.

Analytically, a conic section can be described by a second-degree equation in two variables: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. The type is determined by the discriminant B^2 − 4AC: ellipse (including circle) if negative, parabola if zero, and hyperbola if positive. Standard forms for these curves include the circle (x − h)^2 + (y − k)^2 = r^2, the ellipse (x − h)^2/a^2 + (y − k)^2/b^2 = 1, the parabola y = ax^2 + bx + c, and the hyperbola x^2/a^2 − y^2/b^2 = 1.

Applications of conic sections span astronomy, optics, engineering, and architecture. Orbits of planets are modeled as ellipses (with the Sun at one focus); parabolic reflectors focus waves; and conics appear in tracking, design, and computer graphics. The study of conics has historical roots in classical geometry and continues to influence modern science and technology.