# Semi-major and semi-minor axes

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 widest points of the perimeter. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. For the special case of a circle, the semi-major axis is the radius.

The length of the semi-major axis ${\displaystyle a}$ of an ellipse is related to the semi-minor axis's length ${\displaystyle b}$ through the eccentricity ${\displaystyle e}$ and the semi-latus rectum ${\displaystyle \ell }$, as follows:

The semi-major 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 ${\displaystyle \ell }$ fixed. Thus ${\displaystyle a}$ and ${\displaystyle b}$ tend to infinity, ${\displaystyle a}$ faster than ${\displaystyle b}$.

The semi-minor axis of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section.

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.

## Related Topics

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