## ::Semi-major axis

### ::concepts

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. Essentially, it is the radius of an orbit at the orbit's two most distant points. For the special case of a circle, the semi-major axis is the radius. One can think of the semi-major axis as an ellipse's *long radius*.

The length of the semi-major axis *a* of an ellipse is related to the semi-minor axis's length *b* through the eccentricity *e* and the semi-latus rectum *ℓ*, as follows:

- <math>b = a \sqrt{1-e^2},\,</math>
- <math>\ell=a(1-e^2),\,</math>
- <math>a\ell=b^2.\,</math>

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 (turning point) 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 *ℓ* fixed. Thus <math>a\,\!</math> and <math>b\,\!</math> tend to infinity, *a* faster than *b*.

**Semi-major axis sections**

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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. Essentially, it is the radius of an orbit at the orbit's two most distant points. For the special case of a circle, the semi-major axis is the radius. One can think of the semi-major axis as an ellipse's *long radius*.

The length of the semi-major axis *a* of an ellipse is related to the semi-minor axis's length *b* through the eccentricity *e* and the semi-latus rectum *ℓ*, as follows:

- <math>b = a \sqrt{1-e^2},\,</math>
- <math>\ell=a(1-e^2),\,</math>
- <math>a\ell=b^2.\,</math>

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 (turning point) 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 *ℓ* fixed. Thus <math>a\,\!</math> and <math>b\,\!</math> tend to infinity, *a* faster than *b*.

**Semi-major axis sections**

Intro Ellipse Hyperbola Astronomy References External links

PREVIOUS: Intro | NEXT: Ellipse |

<< | >> |