# Semi-major and semi-minor axes

The semi-major (a) and semi-minor axis (b) of an ellipse

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. For the special case of a circle, the semi-major axis is the radius.

The length of the semi-major axis ${\displaystyle a}$ of an ellipse is related to the semi-minor axis's length ${\displaystyle b}$ through the eccentricity ${\displaystyle e}$ and the semi-latus rectum ${\displaystyle \ell }$, as follows:

{\displaystyle {\begin{aligned}b&=a{\sqrt {1-e^{2}}},\,\\\ell &=a\left(1-e^{2}\right),\,\\a\ell &=b^{2}.\,\end{aligned}}}

The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches. Thus it is the distance from the center to either vertex of the hyperbola.

A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping ${\displaystyle \ell }$ fixed. Thus ${\displaystyle a}$ and ${\displaystyle b}$ tend to infinity, ${\displaystyle a}$ faster than ${\displaystyle b}$.

The semi-minor axis of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section.

The major and minor axes are the axes of symmetry for the curve: in an ellipse, the minor axis is the shorter one; in a hyperbola, it is the one that does not intersect the hyperbola.

## Ellipse

The equation of an ellipse is:

${\displaystyle {\frac {\left(x-h\right)^{2}}{a^{2}}}+{\frac {\left(y-k\right)^{2}}{b^{2}}}=1.}$

Where (h,k) is the center of the ellipse in Cartesian coordinates, in which an arbitrary point is given by (x,y).

The semi-major axis is the mean value of the maximum and minimum distances ${\displaystyle r_{\max }}$ and ${\displaystyle r_{\min }}$ of the ellipse from a focus — that is, of the distances from a focus to the endpoints of the major axis. In astronomy these extreme points are called apsides.

${\displaystyle a={\frac {r_{\max }+r_{\min }}{2}}.}$

The semi-minor axis of an ellipse is the geometric mean of these distances:

${\displaystyle b={\sqrt {r_{\max }r_{\min }}}.}$

The eccentricity of an ellipse is defined as

${\displaystyle e={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}}$ so ${\displaystyle r_{\min }=a(1-e),r_{\max }=a(1+e)}$.

Now consider the equation in polar coordinates, with one focus at the origin and the other on the ${\displaystyle (\theta =\pi )-}$direction,

${\displaystyle r(1+e\cos \theta )=\ell .\,}$

The mean value of ${\displaystyle r=\ell /(1-e)}$ and ${\displaystyle r=\ell /(1+e)}$, for ${\displaystyle \theta =\pi }$ and ${\displaystyle \theta =0}$ is

${\displaystyle a={\ell \over 1-e^{2}}.\,}$

In an ellipse, the semi-major axis is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix.

The semi-minor axis of an ellipse runs from the center of the ellipse (a point halfway between and on the line running between the foci) to the edge of the ellipse. The semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the major axis that connects two points on the ellipse's edge.

The semi-minor axis ${\displaystyle b}$ is related to the semi-major axis ${\displaystyle a}$ through the eccentricity ${\displaystyle e}$ and the semi-latus rectum ${\displaystyle \ell }$, as follows:

{\displaystyle {\begin{aligned}b&=a{\sqrt {1-e^{2}}}\,\!\\a\ell &=b^{2}.\,\!\end{aligned}}}

A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping ${\displaystyle \ell }$ fixed. Thus ${\displaystyle a}$ and ${\displaystyle b}$ tend to infinity, ${\displaystyle a}$ faster than ${\displaystyle b}$.

The length of the semi-minor axis could also be found using the following formula,[1]

${\displaystyle 2b={\sqrt {(p+q)^{2}-f^{2}}}}$

where ${\displaystyle f}$ is the distance between the foci, ${\displaystyle p}$ and ${\displaystyle q}$ are the distances from each focus to any point in the ellipse.

Other Languages
asturianu: Semiexe mayor
বাংলা: পরাক্ষ
беларуская: Вялікая паўвось
български: Голяма полуос
català: Semieix major
español: Semieje mayor
Esperanto: Granda duonakso
français: Grand axe
한국어: 긴반지름
Bahasa Indonesia: Sumbu semi-mayor
македонски: Голема полуоска
Bahasa Melayu: Paksi semimajor

norsk nynorsk: Stor halvakse
occitan: Semiaxe major
português: Semieixo maior
română: Semiaxa mare
Simple English: Semi-major axis
slovenčina: Veľká polos
српски / srpski: Велика полуоса
srpskohrvatski / српскохрватски: Velika poluosa
svenska: Halv storaxel
Türkçe: Ana eksen
українська: Велика піввісь
Tiếng Việt: Bán trục lớn