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In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: The eccentricity of a circle is 0. The eccentricity of an ellipse which is not a circle is between 0 and 1.
The constant is given by =, where e is the eccentricity of the conic section. The equation for a conic section with apex at the origin and tangent to the y axis is y 2 − 2 R x + ( K + 1 ) x 2 = 0 {\displaystyle y^{2}-2Rx+(K+1)x^{2}=0}
Conic sections of varying eccentricity sharing a focus point and directrix line, including an ellipse (red, e = 1/2), a parabola (green, e = 1), and a hyperbola (blue, e = 2). The conic of eccentricity 0 in this figure is an infinitesimal circle centered at the focus, and the conic of eccentricity ∞ is an infinitesimally separated pair of lines.
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.
The basic idea behind geometric continuity was that the five conic sections were really five different versions of the same shape. An ellipse tends to a circle as the eccentricity approaches zero, or to a parabola as it approaches one; and a hyperbola tends to a parabola as the eccentricity drops toward one; it can also tend to intersecting lines.
A pencil of confocal ellipses and hyperbolas is specified by choice of linear eccentricity c (the x-coordinate of one focus) and can be parametrized by the semi-major axis a (the x-coordinate of the intersection of a specific conic in the pencil and the x-axis). When 0 < a < c the conic is a hyperbola; when c < a the conic is an ellipse.
Neptune's and Venus's have even lower eccentricities of 0.008 6 and 0.006 8 respectively, the latter being the least orbital eccentricity of any planet in the Solar System. Over hundreds of thousands of years, the eccentricity of the Earth's orbit varies from nearly 0.003 4 to almost 0.058 as a result of gravitational attractions among the planets.
whose values for =,,,, are ,,,, – there are no curves of degree 0 (a single point is a point and is thus determined by a point, which is codimension 2), 2 points determine a line, 5 points determine a conic, 9 points determine a cubic, 14 points determine a quartic, and so forth.