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Semi-minor axis

The semi minor axis of an ellipse

In geometry, the semi-minor axis (also semiminor axis) is a line segment associated with most conic sections (that is, with ellipses and hyperbolas). One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis. It is one of the axes of symmetry for the curve: in an ellipse, the shorter one; in a hyperbola, the one that does not intersect the hyperbola.

Ellipse

The semi-minor axis of an ellipse is one half of the minor axis, running from the center, halfway between and perpendicular to the line running between the foci, and to the edge of the ellipse. The minor axis is the longest line segment that runs perpendicular to the major axis.

The semi-minor axis b is related to the semi-major axis $a$ through the eccentricity $e$ and the semi-latus rectum $l$ , as follows:

$b = a \sqrt{1-e^2}\,\!$

$al=b^2\,\!$ .

The semi-minor axis of an ellipse is the geometric mean of the maximum and minimum distances $r_{max}$ and $r_{min}$ of the ellipse from a focus — that is, of the distances from a focus to the endpoints of the major axis:

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

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 l fixed. Thus a and b tend to infinity, a faster than b.

Hyperbola

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, ±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 semi-minor axis, of length b. Denoting the semi-major axis length (distance from the center to a vertex) as a, the semi-minor and semi-major axes' lengths appear in the equation of the hyperbola relative to these axes as follows:

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.$

The semi-minor axis and the semi-major axis are related through the eccentricity, as follows:

$b = a \sqrt{e^2-1}.$

Note that in a hyperbola b can be larger than a. [1]

External links

Semi-minor and semi-major axes of an ellipse With interactive animation

Articles related to orbits

Orbits

General	Box Capture Circular Elliptical / Highly elliptical Escape Graveyard Hyperbolic trajectory Inclined / Non-inclined Osculating Parabolic trajectory Parking Synchronous semi sub

Geocentric	Geosynchronous Geostationary Sun-synchronous Low Earth Medium Earth High Earth Molniya Near-equatorial Orbit of the Moon Polar Tundra Two-line elements

About other points	Areosynchronous Areostationary Halo Lissajous Lunar Heliocentric Heliosynchronous

Parameters

Shape/Size	$e\,\!$ Eccentricity $a\,\!$ Semi-major axis $b\,\!$ Semi-minor axis $Q,q\,\!$ Apsides

Orientation	$i\,\!$ Inclination $\Omega\,\!$ Longitude of the ascending node $\omega\,\!$ Argument of periapsis $\varpi\,\!$ Longitude of the periapsis

Position	$M\,\!$ Mean anomaly $\nu\,\!$ True anomaly $E\,\!$ Eccentric anomaly $L\,\!$ Mean longitude $l\,\!$ True longitude

Variation	$T\,\!$ Orbital period $n\,\!$ Mean motion $v\,\!$ Orbital speed $t_0\,\!$ Epoch

Maneuvers

Collision avoidance (spacecraft) Delta-v Delta-v budget Bi-elliptic transfer Geostationary transfer Gravity assist Gravity turn Hohmann transfer Low energy transfer Oberth effect Inclination change Phasing Rocket equation Rendezvous Transposition, docking, and extraction

Other orbital mechanics topics

Celestial coordinate system Characteristic energy Ephemeris Equatorial coordinate system Ground track Hill sphere Interplanetary Transport Network Kepler's laws of planetary motion Lagrangian point n-body problem Orbit equation Orbital state vectors Perturbation Retrograde motion Specific orbital energy Specific relative angular momentum

List of orbits

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