Eccentricity (math)
The term eccentricity has two related meanings in mathematics in connection with non-degenerate conic sections (ellipses, hyperbolas, parabolas):
- 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 .
- The numerical eccentricity for ellipses and hyperbolas is the ratio of the linear eccentricity to the major semi-axis and thus a dimensionless number.
- The following applies to an ellipse . In the case , the ellipse is a circle .
- The numerical eccentricity describes here the increasing deviation of an ellipse from the circular shape.
- 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 .
- For a parabola one defines (for motivation see below).
- 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.
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.
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.
Mathematical treatment
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.
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
- (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).
- The parameter only exists in the case of ellipses and hyperbolas and is called linear eccentricity . is a length.
- For the ellipse is .
- For is and the ellipse a circle. Is only a little smaller than , i.e. H. is small, then the ellipse is very flat.
- For the hyperbola is and therefore for every hyperbola .
- The parameter exists for ellipses, hyperbolas and parabolas and is called numerical eccentricity . is the ratio of two lengths, so it is dimensionless .
- For ellipses and hyperbolas applies , for parabolas .
If one understands an ellipse / parabola / hyperbola as a plane section of a vertical circular cone, the numerical eccentricity can be explained
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.)
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) ).
Web links
- Eric W. Weisstein : Eccentricity . In: MathWorld (English).
Individual evidence
- ^ Jacob Steiner's lectures on synthetic geometry. BG Teubner, Leipzig 1867 (at Google Books: books.google.de ).
- ^ Graf, Barner: Descriptive Geometry. Quelle & Meyer-Verlag, 1973, pp. 169-173.