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How does eccentricity distinguish a circle, ellipse, parabola, and hyperbola?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

In analytic geometry, eccentricity is a constant that defines the shape of a conic section. It represents the ratio of the distance from any point on the curve to a fixed point (the focus) to the distance from that same point to a fixed line (the directrix).12

The value of the eccentricity distinguishes the four types of conic sections as follows:

Circle: The eccentricity is exactly 0. A circle is a special case of an ellipse where the distance between the foci is zero.134
Ellipse: The eccentricity is greater than 0 but less than 1. It measures the degree of "flattening" of the curve; as the eccentricity approaches zero, the ellipse becomes more circular.134
Parabola: The eccentricity is exactly 1.1
Hyperbola: The eccentricity is greater than 1.1

By determining this ratio, mathematicians can classify any conic section and understand its geometric properties.1