conischen

Conic sections, or conics, are curves obtained by intersecting a plane with a right circular cone. The position and angle of the plane determine whether the intersection is a circle, an ellipse, a parabola, or a hyperbola. The circle is a special case of an ellipse where the two axes are equal.

Ellipses arise when the plane cuts through the cone at an angle to the axis without passing through the base, resulting in a closed curve with two focal points. The sum of the distances from any point on an ellipse to the two foci remains constant. Parabolas occur when the plane is parallel to a generating line of the cone; they have a single focus and a directrix, with all points equidistant from the focus and the directrix. Hyperbolas are formed when the plane cuts both nappes of the cone, producing two separate branches and a constant difference of distances to two foci.

Equations of conics are central to analytic geometry. A standard ellipse has the form x^2/a^2 + y^2/b^2 = 1, with a ≠ b; the circle is the special case a = b. A parabola can be written as y^2 = 4ax (opening along the x-axis). A hyperbola has the form x^2/a^2 - y^2/b^2 = 1. The eccentricity e distinguishes the types: e = 0 for a circle; 0 < e < 1 for an ellipse; e = 1 for a parabola; e > 1 for a hyperbola.

Conics appear across mathematics and science, from the description of planetary orbits and optical paths to architectural design and computer graphics. They also serve as a foundational concept in classical geometry and algebra.