Semi-minor axis
The semi-major (in red*) and semi-minor axis (in blue*) of an ellipse.
(* on some browsers)
In
geometry, the
semi-minor axis (also
semiminor axis) is a
line segment associated with most
conic sections (that is, with
ellipses and
hyperbolas) that is at
right angles with the
semi-major axis and has one end at the center of the conic section. 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 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
b is related to the
semi-major axis through the
eccentricity and the
semi-latus rectum , as follows:
- .
The semi-minor axis of an ellipse is the
geometric mean of the maximum and minimum distances
and
of the ellipse from a focus — that is, of the distances from a focus to the endpoints of the major axis:
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.
The length of the semi-minor axis could also be found using the following formula,
[1]
- where f is the distance between the foci, p and q are the distances from each focus to any point in the ellipse.
Hyperbola
In a hyperbola, a
conjugate axis or
minor axis of length 2
b, 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:
The semi-minor axis and the semi-major axis are related through the eccentricity, as follows:
Note that in a hyperbola
b can be larger than
a.
[1]
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