ellipse: Quick Facts

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Eccentricity of conic sectionsThe eccentricity of a conic section completely characterizes its shape. For example, all circles have zero eccentricity, and all parabolas have unit eccentricity; hence, all circles (and all parabolas) have the same shape, only varying in size. (Under appropriate magnification they are indistinguishable.) In contrast, ellipses and hyperbolas vary greatly in shape.

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.

A diagram of a double cone intersected by two green planes labeled PP (picture plane) and RP (reality plane), illustrating projected conic sections in projective geometry, with the apex and a point Omega marked.

Figure 5: Ellipse and the major (or auxiliary) circle. The two foci of the ellipse are F and F′. A line that is drawn perpendicular to the x-axis at S cuts the ellipse and the circle at P and P′, respectively.

A three-dimensional graph of an elliptic paraboloid surface, shaped like an upward-opening bowl, with grid lines and labeled x, y, and z axes. The surface curves upward from the origin, illustrating the equation z = x squared plus y squared.

Figure 4: The standard form of the ellipse (see text).

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eccentric circle minor axis major axis semimajor axis semiminor axis Apse conic section parabola hyperbola circle

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