the hyperbola, is the length of the semi-major axis and is the length of the semi-minor axis. When the conic section is given in the general quadratic form the following formula gives the eccentricity if the conic section is not a parabola (which has eccentricity equal to 1), not a degenerate hyperbola or degenerate ellipse, and not an imaginary ellipse: where if the determinant of the 3×3 matrix is negative or if that determinant is positive. Ellipses The eccentricity of an ellipse is strictly less than 1. When circles (which have eccentricity 0) are counted as ellipses, the eccentricity of an ellipse is greater than or equal to 0; if circles are given a special category and are excluded from the category of ellipses, then the eccentricity of an ellipse is strictly greater than 0. For any ellipse, let be the length of its semi-major axis and be the length of its semi-minor axis. We define a number of related additional concepts (only for ellipses): Other formulae for the eccentricity of an ellipse The eccentricity of an ellipse is, most simply, the ratio of the distance between the center of the ellipse and each focus to the length of the semimajor axis . The eccentricity is also the ratio of the semimajor axis to the distance from the center to the directrix: The eccentricity can be expressed in terms of the flattening (defined as for semimajor axis and semiminor axis ): (Flattening may be denoted by in some subject areas if is linear eccentricity.) Define the maximum and minimum radii and as the maximum and minimum distances from either focus to the ellipse (that is, the distances from either focus to the two ends of the major axis). Then with semimajor axis , the eccentricity is given by which is the distance between the foci divided by the length of the major axis. Hyperbolas The eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of a rectangular hyperbola is . Quadrics The eccentricity of a three-dimensional quadric is the eccentricity of a designated section of it. 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