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 two most widely separated points of the perimeter. The semimajor axis ( major ) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semiminor axis ( minor semiaxis) of an ellipse or hyperbola is a line segment that is at with the semimajor axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semiaxes are both equal to the radius of the circle.
The length of the semimajor axis of an ellipse is related to the semiminor axis's length through the eccentricity and the semilatus rectum $\backslash ell$, as follows:
The semimajor 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 $\backslash ell$ fixed. Thus and tend to infinity, faster than .
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.
where ( h, k) is the center of the ellipse in Cartesian coordinates, in which an arbitrary point is given by ( x, y).
The semimajor axis is the mean value of the maximum and minimum distances $r\_\backslash text\{max\}$ and $r\_\backslash text\{min\}$ 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 apsis.
The semiminor axis of an ellipse is the geometric mean of these distances:
The eccentricity of an ellipse is defined as so
Now consider the equation in polar coordinates, with one focus at the origin and the other on the $\backslash theta\; =\; \backslash pi$ direction:
The mean value of $r\; =\; \backslash ell\; /\; (1\; \; e)$ and $r\; =\; \backslash ell\; /\; (1\; +\; e)$, for $\backslash theta\; =\; \backslash pi$ and $\backslash theta\; =\; 0$ is
In an ellipse, the semimajor axis is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix.
The semiminor 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 semiminor 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 semiminor axis is related to the semimajor axis through the eccentricity and the semilatus rectum $\backslash ell$, as follows:
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 $\backslash ell$ fixed. Thus and tend to infinity, faster than .
The length of the semiminor axis could also be found using the following formula: "Major / Minor axis of an ellipse", Math Open Reference, 12 May 2013.
where is the distance between the foci, and are the distances from each focus to any point in the ellipse.
In terms of the semilatus rectum and the eccentricity we have
The transverse axis of a hyperbola coincides with the major axis.
In a hyperbola, a conjugate axis or minor axis of length $2b$, corresponding to the minor axis of an ellipse, can be drawn perpendicular to the transverse axis or major axis, the latter connecting the two vertices (turning points) of the hyperbola, with the two axes intersecting at the center of the hyperbola. The endpoints $(0,\backslash pm\; b)$ of the minor axis lie at the height of the asymptotes over/under the hyperbola's vertices. Either half of the minor axis is called the semiminor axis, of length . Denoting the semimajor axis length (distance from the center to a vertex) as , the semiminor and semimajor axes' lengths appear in the equation of the hyperbola relative to these axes as follows:
The semiminor axis is also the distance from one of focuses of the hyperbola to an asymptote. Often called the impact parameter, this is important in physics and astronomy, and measure the distance a particle will miss the focus by if its journey is unperturbed by the body at the focus.
The semiminor axis and the semimajor axis are related through the eccentricity, as follows:
Note that in a hyperbola can be larger than .
where:
Note that for all ellipses with a given semimajor axis, the orbital period is the same, disregarding their eccentricity.
The specific angular momentum of a small body orbiting a central body in a circular or elliptical orbit is
where:
In astronomy, the semimajor axis is one of the most important orbital elements of an orbit, along with its orbital period. For Solar System objects, the semimajor axis is related to the period of the orbit by Kepler's third law (originally derived):
where is the period, and is the semimajor axis. This form turns out to be a simplification of the general form for the twobody problem, as determined by Isaac Newton:
where is the gravitational constant, is the mass of the central body, and is the mass of the orbiting body. Typically, the central body's mass is so much greater than the orbiting body's, that may be ignored. Making that assumption and using typical astronomy units results in the simpler form Kepler discovered.
The timeaveraged value of the reciprocal of the radius, $r^\{1\}$, is $a^\{1\}$.
for an elliptical orbit and, depending on the convention, the same or
for a hyperbolic trajectory, and
(specific orbital energy) and
(standard gravitational parameter), where:
Note that for a given amount of total mass, the specific energy and the semimajor axis are always the same, regardless of eccentricity or the ratio of the masses. Conversely, for a given total mass and semimajor axis, the total specific orbital energy is always the same. This statement will always be true under any given conditions.
The reason for the assumption of prominent elliptical orbits lies probably in the much larger difference between aphelion and perihelion. That difference (or ratio) is also based on the eccentricity and is computed as $\backslash frac\{r\_\backslash text\{a\}\}\{r\_\backslash text\{p\}\}\; =\; \backslash frac\{1\; +\; e\}\{1\; \; e\}$. Due to the large difference between aphelion and perihelion, Kepler's second law is easily visualized.

