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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.
Further, the orthogonal trajectories of these ellipses comprise the elliptic curves with j ≤ 1, and any ellipse in described as a locus relative to two foci is uniquely the elliptic curve sum of two Steiner ellipses, obtained by adding the pairs of intersections on each orthogonal trajectory. Here, the vertex of the hyperboloid serves as the ...
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.
Fitting of a noisy curve by an asymmetrical peak model, with an iterative process (Gauss–Newton algorithm with variable damping factor α).Curve fitting [1] [2] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, [3] possibly subject to constraints.
Computational methods are available for generating pseudo-random vectors from elliptical distributions, for use in Monte Carlo simulations for example. [3] Some elliptical distributions are alternatively defined in terms of their density functions. An elliptical distribution with a density function f has the form:
The semi-minor axis of an ellipse runs from the center of the ellipse (a point halfway between and on the line running between the foci) to the edge of the ellipse. The semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the major axis that connects two points on the ellipse's edge.
For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in ...
Those integrals are in turn named elliptic because they first were encountered for the calculation of the arc length of an ellipse. Important elliptic functions are Jacobi elliptic functions and the Weierstrass ℘-function. Further development of this theory led to hyperelliptic functions and modular forms.
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