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