Namespaces
Variants
Actions

Difference between revisions of "Eccentricity"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
The number equal to the ratio of the distance between any point of a conic section (cf. [[Conic sections|Conic sections]]) and a given point (the focus) to that between the same point and a given line (the directrix). Two conic sections having the same eccentricity are similar. For an ellipse the eccentricity is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e0350301.png" /> (for a circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e0350302.png" />), for a hyperbola <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e0350303.png" />, and for a parabola <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e0350304.png" />. For an ellipse and hyperbola the eccentricity can also be defined as the ratio of the distances between the foci and the length of the major axis.
+
<!--
 +
e0350301.png
 +
$#A+1 = 19 n = 0
 +
$#C+1 = 19 : ~/encyclopedia/old_files/data/E035/E.0305030 Eccentricity
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
 +
{{TEX|auto}}
 +
{{TEX|done}}
  
 +
The number equal to the ratio of the distance between any point of a conic section (cf. [[Conic sections|Conic sections]]) and a given point (the focus) to that between the same point and a given line (the directrix). Two conic sections having the same eccentricity are similar. For an ellipse the eccentricity is  $  e < 1 $(
 +
for a circle  $  e = 0 $),
 +
for a hyperbola  $  e > 1 $,
 +
and for a parabola  $  e = 1 $.
 +
For an ellipse and hyperbola the eccentricity can also be defined as the ratio of the distances between the foci and the length of the major axis.
  
 
====Comments====
 
====Comments====
 
The quantity defined in the article above is often called the numerical eccentricity. The linear eccentricity equals half the distance between the foci (cf. [[Focus|Focus]]). See [[#References|[a1]]], Chapt. 17; [[#References|[a2]]], p. 117.
 
The quantity defined in the article above is often called the numerical eccentricity. The linear eccentricity equals half the distance between the foci (cf. [[Focus|Focus]]). See [[#References|[a1]]], Chapt. 17; [[#References|[a2]]], p. 117.
  
For the  "standard"  ellipse, parabola and hyperbola, given, respectively, by the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e0350305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e0350306.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e0350307.png" />, the eccentricity is equal to, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e0350308.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e0350309.png" />), 1 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e03503010.png" />. A focus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e03503011.png" /> and a corresponding directrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e03503012.png" /> for the three cases are given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e03503013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e03503014.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e03503015.png" />); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e03503016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e03503017.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e03503018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035030/e03503019.png" />. There are two foci for ellipses and hyperbolas and there is one for parabolas.
+
For the  "standard"  ellipse, parabola and hyperbola, given, respectively, by the equations $  {x  ^ {2} } / {a  ^ {2} } + {y  ^ {2} } / {b  ^ {2} } = 1 $,  
 +
$  y  ^ {2} = 2px $,  
 +
and $  {x  ^ {2} } / {a  ^ {2} } - {y  ^ {2} } / {b  ^ {2} } = 1 $,  
 +
the eccentricity is equal to, respectively, $  a  ^ {-} 1 \sqrt {a  ^ {2} - b  ^ {2} } $(
 +
if $  a > b $),  
 +
1 and $  a  ^ {-} 1 \sqrt {a  ^ {2} + b  ^ {2} } $.  
 +
A focus $  f $
 +
and a corresponding directrix $  D $
 +
for the three cases are given by $  f = ( \sqrt {a  ^ {2} - b  ^ {2} } , 0) $,
 +
$  D : x = a  ^ {2} ( a  ^ {2} - b  ^ {2} )  ^ {-} 1/2 $(
 +
if $  a > b $);  
 +
$  f = (- p/2 , 0) $,
 +
$  D : x = p/2 $;  
 +
$  f = ( \sqrt {a  ^ {2} + b  ^ {2} } , 0) $,
 +
$  D : x = a  ^ {2} ( a  ^ {2} + b  ^ {2} )  ^ {-} 1/2 $.  
 +
There are two foci for ellipses and hyperbolas and there is one for parabolas.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1963)</TD></TR></table>

Latest revision as of 19:36, 5 June 2020


The number equal to the ratio of the distance between any point of a conic section (cf. Conic sections) and a given point (the focus) to that between the same point and a given line (the directrix). Two conic sections having the same eccentricity are similar. For an ellipse the eccentricity is $ e < 1 $( for a circle $ e = 0 $), for a hyperbola $ e > 1 $, and for a parabola $ e = 1 $. For an ellipse and hyperbola the eccentricity can also be defined as the ratio of the distances between the foci and the length of the major axis.

Comments

The quantity defined in the article above is often called the numerical eccentricity. The linear eccentricity equals half the distance between the foci (cf. Focus). See [a1], Chapt. 17; [a2], p. 117.

For the "standard" ellipse, parabola and hyperbola, given, respectively, by the equations $ {x ^ {2} } / {a ^ {2} } + {y ^ {2} } / {b ^ {2} } = 1 $, $ y ^ {2} = 2px $, and $ {x ^ {2} } / {a ^ {2} } - {y ^ {2} } / {b ^ {2} } = 1 $, the eccentricity is equal to, respectively, $ a ^ {-} 1 \sqrt {a ^ {2} - b ^ {2} } $( if $ a > b $), 1 and $ a ^ {-} 1 \sqrt {a ^ {2} + b ^ {2} } $. A focus $ f $ and a corresponding directrix $ D $ for the three cases are given by $ f = ( \sqrt {a ^ {2} - b ^ {2} } , 0) $, $ D : x = a ^ {2} ( a ^ {2} - b ^ {2} ) ^ {-} 1/2 $( if $ a > b $); $ f = (- p/2 , 0) $, $ D : x = p/2 $; $ f = ( \sqrt {a ^ {2} + b ^ {2} } , 0) $, $ D : x = a ^ {2} ( a ^ {2} + b ^ {2} ) ^ {-} 1/2 $. There are two foci for ellipses and hyperbolas and there is one for parabolas.

References

[a1] M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a2] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
How to Cite This Entry:
Eccentricity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Eccentricity&oldid=46789
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article