ellipsís

Ellipsís is the set of points in a plane such that the sum of the distances to two fixed points, called the foci, is constant. It is one of the conic sections, arising when a plane intersects a double cone at an angle that cuts both nappes but does not pass through the apex.

Geometric properties are central to the ellipsís. In a Cartesian system with the center at the origin and the major axis on the x-axis, its standard equation is x^2/a^2 + y^2/b^2 = 1, where a ≥ b. The foci lie at (±c, 0), with c^2 = a^2 − b^2. The eccentricity e = c/a satisfies 0 < e < 1. The lengths of the major and minor axes are 2a and 2b, respectively.

A common way to describe the ellipsís is by parametric equations: x = a cos t, y = b sin t, for t in [0, 2π]. If the figure is rotated, the general second-degree form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 with B^2 − 4AC < 0 describes any orientation.

Key properties include symmetry about both principal axes and the focal definition. The area is πab, while there is no simple exact formula for the perimeter; practical approximations exist, such as Ramí’s formula. In the circle limit (a = b), the ellipsís becomes a circle. The ellipsís appears in physics and engineering, notably in planetary orbits and optical reflections.