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Semi-minor axis

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Title: Semi-minor axis  
Author: World Heritage Encyclopedia
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Subject: Semi-major axis, Eccentricity (mathematics), Apsis, Kepler's laws of planetary motion, Hyperbola
Collection: Conic Sections, Orbits
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Semi-minor axis

The semi-major axis (in red) and semi-minor axis (in blue) 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) 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.

Contents

  • Ellipse 1
  • Hyperbola 2
  • References 3
  • External links 4

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 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.

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

2b = \sqrt{(p+q)^2 -f^2} 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 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. [2]

References

  1. ^ http://www.mathopenref.com/ellipseaxes.html,"Major / Minor axis of an ellipse",Math Open Reference, 12 May 2013

External links

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