elipse

An elipse, also known as an ellipse in English, is a planar curve with several equivalent definitions. One common definition is the locus of all points for which the sum of the distances to two fixed points, called the foci, is constant. Another is that it is a conic section formed when a plane cuts through a cone at an angle shallower than the cone’s side. The elipse is a closed, smooth curve with eccentricity e less than 1.

Geometric properties and equations. A centered elipse has major axis length 2a and minor axis length 2b, with a ≥ b. The distance from the center to each focus is c, where c^2 = a^2 − b^2, and the eccentricity is e = c/a = sqrt(1 − (b^2/a^2)). In standard position with its axes aligned to the coordinate axes, the elipse is described by (x − h)^2/a^2 + (y − k)^2/b^2 = 1, where (h, k) is the center. If the ellipse is rotated, a general quadratic form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 with B^2 < 4AC applies. Foci lie along the major axis at (h ± c, k).

Area and perimeter. The area is A = πab. There is no simple exact formula for the perimeter; it is commonly approximated, for example by Ramanujan’s formula P ≈ π[3(a + b) − sqrt((3a + b)(a + 3b))].

Coordinate representations and parametric form. A common parametric representation is x = h + a cos t, y = k + b sin t, with t in [0, 2π). In polar form, with a focus at the origin and major axis along the polar axis, r = l/(1 − e cos θ), where l = b^2/a is the semi-latus rectum.

Uses and history. Ellipses appear in astronomy (planetary orbits with the Sun at a focus), optics (elliptical reflectors and caustics), mechanics, and design. The concept arises from the study of conic sections by the ancient Greeks, with the term ellipse popularized in the early modern period; the name derives from Greek meaning “omission” or “defect.”