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The linear eccentricity of an ellipse or hyperbola, denoted c (or sometimes f or e), is the distance between its center and either of its two foci. The eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a : that is, e = c a {\displaystyle e={\frac {c}{a}}} (lacking a center, the linear eccentricity for ...
In an ellipse, the semi-major axis is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix. 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 ...
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.
A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a fixed positive constant, called the eccentricity e. If 0 < e < 1 the conic is an ellipse, if e = 1 the conic is a parabola, and if e > 1 the conic is a hyperbola.
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 points of tangency F 1, F 2 are the foci of the blue ellipse. The spheres are also tangent to the cone at circles k 1 , k 2 . For a point P on the ellipse, the tangent segments PF 1 and PF 2 can each be reflected to other tangents of equal length, PF 1 = PP 1 and PF 2 = PP 2 , with PP 1 and PP 2 colinear along the ray SP.
A peripheral vertex in a graph of diameter d is one whose eccentricity is d —that is, a vertex whose distance from its furthest vertex is equal to the diameter. Formally, v is peripheral if ϵ(v) = d. A pseudo-peripheral vertex v has the property that, for any vertex u, if u is as far away from v as possible, then v is as far away from u as
The eccentric anomaly E is one of the angles of a right triangle with one vertex at the center of the ellipse, its adjacent side lying on the major axis, having hypotenuse a (equal to the semi-major axis of the ellipse), and opposite side (perpendicular to the major axis and touching the point P′ on the auxiliary circle of radius a) that ...