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A circle of radius a compressed to an ellipse. A sphere of radius a compressed to an oblate ellipsoid of revolution. Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution respectively. Other terms used are ellipticity, or oblateness.
The eccentricity of a circle is 0. The eccentricity of an ellipse which is not a circle is between 0 and 1. The eccentricity of a parabola is 1. The eccentricity of a hyperbola is greater than 1. The eccentricity of a pair of lines is . Two conic sections with the same eccentricity are similar.
An ellipse (red) obtained as the intersection of a cone with an inclined plane. Ellipse: notations Ellipses: examples with increasing eccentricity. In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant.
An ellipse has two axes and two foci Unlike most other elementary shapes, such as the circle and square , there is no algebraic equation to determine the perimeter of an ellipse . Throughout history, a large number of equations for approximations and estimates have been made for the perimeter of an ellipse.
The solid curve is the ellipse, with m = 1 − 1/b 2 and u = F(φ,m) where F(⋅,⋅) is the elliptic integral of the first kind (with parameter =). The dotted curve is the unit circle. Tangent lines from the circle and ellipse at x = cd crossing the x-axis at dc are shown in light grey.
More generally, for any collection of points P i, weights w i, and constant C, one can define a circle as the locus of points X such that (,) =.. The director circle of an ellipse is a special case of this more general construction with two points P 1 and P 2 at the foci of the ellipse, weights w 1 = w 2 = 1, and C equal to the square of the major axis of the ellipse.
In the 2-dimensional case, if the density exists, each iso-density locus (the set of x 1,x 2 pairs all giving a particular value of ()) is an ellipse or a union of ellipses (hence the name elliptical distribution).
The 1-ellipse is the circle, and the 2-ellipse is the classic ellipse. Both are algebraic curves of degree 2. For any number n of foci, the n-ellipse is a closed, convex curve. [2]: (p. 90) The curve is smooth unless it goes through a focus. [5]: p.7