Eccentricity is a measure of how closely a conic section resembles a circle. It is a characteristic parameter of every conic section and conic sections are said to be similar if and only if their eccentricities are equal. Parabolas and hyperbolas have only one type of eccentricity but ellipses have three. The term "eccentricity" typically refers to the first eccentricity of an ellipse unless otherwise specified. This value also has other names such as "numerical eccentricity" and "half-focal separation" in the case of ellipses and hyperbolas.
Interpret the value of the eccentricity. The eccentricity ranges from 0 to infinity and the greater the eccentricity, the less the conic section resembles a circle. A conic section with an eccentricity of 0 is a circle. An eccentricity less than 1 indicates an ellipse, an eccentricity of 1 indicates a parabola and an eccentricity greater than 1 indicates a hyperbola.
Define some terms. Formulas for eccentricity will represent the eccentricity as e. The length of the semi-major axis will be a and the length of the semi-minor axis will be b.
Evaluate conic sections that have constant eccentricities. Eccentricity may also be defined as e c/a where c is the distance of the focus to the center and a is the length of the semi-major axis. The focus of a circle is its center, so e=0 for all circles. A parabola may be considered to have one focus at infinity, so both the focus and vertices of a parabola are infinitely far from the "center" of the parabola. This makes e=1 for all parabolas.
Find the eccentricity of an ellipse. This is given as e = (1-b^2/a^2)^(1/2). Note that an ellipse with major and minor axes of equal length has an eccentricity of 0 and is therefore a circle. Since a is the length of the semi-major axis, a >= b and therefore 0 <= e < 1 for all ellipses.
Find the eccentricity of a hyperbola. This is given as e = (1+b^2/a^2)^(1/2). Since b^2/a^2 can be any positive value, e may be any value greater than 1.