konisch

Konisch is an adjective in German meaning "conical" or "cone-shaped". In mathematics and related disciplines, it describes objects and curves related to a cone, most notably the conic sections.

Conic sections are the curves obtained by intersecting a plane with a double cone. Depending on the plane’s angle, the intersection is a circle, an ellipse, a parabola, or a hyperbola. A circle is a special case of an ellipse with equal semiaxes.

Key properties include eccentricity e, which measures deviation from a circle: e = 0 for a circle, 0 < e < 1 for an ellipse, e = 1 for a parabola, and e > 1 for a hyperbola. In standard Cartesian form, common conics have simple equations: circle (x−h)^2+(y−k)^2=r^2; ellipse (x−h)^2/a^2+(y−k)^2/b^2=1; parabola (y−k)=a(x−h)^2 or x^2=4p(y−k); hyperbola (x−h)^2/a^2−(y−k)^2/b^2=1.

In projective geometry, a conic is a second‑degree curve in the projective plane, described by a homogeneous quadratic equation. Conics possess invariants and have numerous coordinate representations, depending on the chosen projection or basis.

Applications span science and engineering. In astronomy and orbital mechanics, conic sections describe possible trajectories under an inverse‑square force: bound orbits are ellipses, unbound ones can be parabolic or hyperbolic. Conical shapes also appear in design and manufacturing, such as funnels, nozzles, and tapering components.

Etymology: konisch derives from the Greek konos meaning "cone", via Latin and German usage.