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In mathematics, an ellipse (Greek ???e???? (elleipsis), a 'falling short') is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside. It is the finite or bounded case of a conic section as a shape cut in a cone by a plane, the unbounded cases being the parabola, which like the ellipse remains connected, and the hyperbola, which separates into two connected components or branches. Equivalently an ellipse can be defined as the locus of points, or path traced out, in a plane such that the sum of the distances from the moving point to two fixed points remains constant. The two fixed points are then called foci (singular- focus). When the foci coincide the ellipse becomes a circle and the two distances then coincide as its radius. A variant of this replaces one of the foci with a straight line not passing through the remaining focus, called the directrix; in this case the locus is of a point whose distance from the remaining focus maintains a constant ratio less than one with its distance from the directrix. Yet another definition of an ellipse, the algebraic or implicit definition, is, up to rotation and translation (geometry), any set of points (x,y) in the Cartesian plane satisfying an equation of the form where a and b are any positive real numbers.
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