# Eccentricity (math)

Ellipse with labels
Hyperbola with labels

The term eccentricity has two related meanings in mathematics in connection with non-degenerate conic sections (ellipses, hyperbolas, parabolas):

Circle, ellipse, parabola and hyperbola with numerical eccentricity and the same half parameter (= radius of the circle)
• The linear eccentricity of an ellipse or hyperbola is the distance from a focal point to the center point and is designated by (see figure). It has the dimension of a length. Since a circle is an ellipse with coincident focal points , applies to the circle .${\ displaystyle e}$${\ displaystyle (F_ {1} = F_ {2} = M)}$${\ displaystyle e = 0}$
• The numerical eccentricity for ellipses and hyperbolas is the ratio of the linear eccentricity to the major semi-axis and thus a dimensionless number.${\ displaystyle \ varepsilon}$${\ displaystyle e / a}$
The following applies to an ellipse . In the case , the ellipse is a circle .${\ displaystyle 0 \ leq \ varepsilon <1}$${\ displaystyle \ varepsilon = 0}$
The numerical eccentricity describes here the increasing deviation of an ellipse from the circular shape.${\ displaystyle \ varepsilon}$
The following applies to a hyperbola . As it grows , the hyperbola becomes more and more open, i.e. i.e., the angle between the asymptotes increases . Equilateral hyperbolas, i.e. those with right-angled asymptotes, result for .${\ displaystyle 1 <\ varepsilon}$${\ displaystyle \ varepsilon}$${\ displaystyle \ varepsilon = {\ sqrt {2}}}$
For a parabola one defines (for motivation see below).${\ displaystyle \ varepsilon = 1}$
The significance of the numerical eccentricity results from the fact that every two ellipses or hyperbolas are exactly similar if they have the same numerical eccentricity. Two parabolas ( ) are always similar.${\ displaystyle \ varepsilon \ equiv 1}$

In the case of ellipses and hyperbolas, the distance between the focal points and the center is also called the focal length. In the case of a parabola, however, the distance between the focal point and the vertex is called the focal length. ${\ displaystyle e}$

In astronomy , mostly only numerical eccentricity is used and is simply called eccentricity , but in contrast to the notation in mathematics, it is often referred to as. ${\ displaystyle e}$

## Mathematical treatment

Ellipse with guidelines
To define the numerical eccentricity on the cone

With eccentricity is first described the deviation of an ellipse from the circular shape. The distance from a focal point to the center point was used as a measure of this deviation (see 1st picture). For you get a circle. Since a hyperbola also has a center point and focal points, the term was extended to the hyperbolic case, although one cannot speak of the proximity of a hyperbola to a circle here. A parabola does not have a center point and therefore initially has no eccentricity. ${\ displaystyle e}$${\ displaystyle e = 0}$

Another way of describing the deviation of an ellipse from a circle is the ratio . It is . Again, you get for a circle. In this case , the parameter is also the relationship between the distance of an ellipse point to the focal point and the distance to a guideline , which is used to define the guideline for an ellipse (see 4th picture). (A circle cannot be defined with the help of a guideline.) If the guideline definition also allows values ​​equal to or greater than 1, the curve obtained is a parabola if the ratio is, and hyperbolas in the case . The parameter allows to describe ellipses, parabolas and hyperbolas with a common family parameter. For example describes the equation ${\ displaystyle \ varepsilon = e / a}$${\ displaystyle 0 \ leq \ varepsilon <1}$${\ displaystyle \ varepsilon = 0}$${\ displaystyle \ varepsilon> 0}$${\ displaystyle \ varepsilon}$${\ displaystyle \ varepsilon}$${\ displaystyle \ varepsilon = 1}$${\ displaystyle \ varepsilon> 1}$${\ displaystyle \ varepsilon}$

${\ displaystyle x ^ {2} (\ varepsilon ^ {2} -1) + 2px-y ^ {2} = 0, \ quad \ varepsilon \ geq 0, \ p> 0}$   (see 3rd picture)

all ellipses (incl. circle), the parabola and all hyperbolas that have the zero point as a common vertex, the x-axis as a common axis and the same half parameter (see 1st picture). ( is also the common radius of curvature in the common vertex, see ellipse, parabola, hyperbola). ${\ displaystyle p}$${\ displaystyle p}$

• The parameter only exists in the case of ellipses and hyperbolas and is called linear eccentricity . is a length.${\ displaystyle e}$${\ displaystyle e}$
For the ellipse is .${\ displaystyle {\ tfrac {x ^ {2}} {a ^ {2}}} + {\ tfrac {y ^ {2}} {b ^ {2}}} = 1}$${\ displaystyle e = {\ sqrt {a ^ {2} -b ^ {2}}}
For is and the ellipse a circle. Is only a little smaller than , i.e. H. is small, then the ellipse is very flat.${\ displaystyle a = b}$${\ displaystyle e = 0}$${\ displaystyle e}$${\ displaystyle a}$${\ displaystyle b}$
For the hyperbola is and therefore for every hyperbola .${\ displaystyle {\ tfrac {x ^ {2}} {a ^ {2}}} - {\ tfrac {y ^ {2}} {b ^ {2}}} = 1}$${\ displaystyle e = {\ sqrt {a ^ {2} + b ^ {2}}}}$${\ displaystyle e> a}$
• The parameter exists for ellipses, hyperbolas and parabolas and is called numerical eccentricity . is the ratio of two lengths, so it is dimensionless .${\ displaystyle \ varepsilon}$${\ displaystyle \ varepsilon}$
For ellipses and hyperbolas applies , for parabolas .${\ displaystyle \ varepsilon = e / a = {\ sqrt {1 \ mp {\ frac {b ^ {2}} {a ^ {2}}}}} \ neq 1}$${\ displaystyle \ varepsilon = 1}$

If one understands an ellipse / parabola / hyperbola as a plane section of a vertical circular cone, the numerical eccentricity can be explained

• ${\ displaystyle \ varepsilon = {\ frac {\ sin \ beta} {\ sin \ alpha}}, \ \ 0 <\ alpha <90 ^ {\ circ}, \ 0 \ leq \ beta \ leq 90 ^ {\ circ }}$

express. Here is the angle of inclination of a cone-generating plane and the angle of inclination of the intersecting plane (see picture). For there are circles and for parabolas. (The plane must not contain the apex of the cone.) ${\ displaystyle \ alpha}$${\ displaystyle \ beta}$${\ displaystyle \ beta = 0}$${\ displaystyle \ beta = \ alpha}$

## literature

• Small encyclopedia of mathematics . Verlag Harri Deutsch, 1977, ISBN 3-87144-323-9 , pp. 192, 195, 328, 330.
• Ayoub B. Ayoub: The Eccentricity of a Conic Section. In: The College Mathematics Journal , Vol. 34, No. 2 (March 2003), pp. 116-121 ( JSTOR 3595784 ).
• Ilka Agricola, Thomas Friedrich: Elementary Geometry. AMS, 2008, ISBN 978-0-8218-9067-7 , pp. 63-70 ( excerpt (Google) ).
• Hans-Jochen Bartsch: Pocket book of mathematical formulas for engineers and natural scientists. Hanser, 2014, ISBN 978-3-446-43735-7 , pp. 287-289 ( excerpt (Google) ).