# Semi-axes of the ellipse

Parameters of an ellipse:
 S 1 , S 2 Main parting S 3 , S 4 Secondary parting ${\ displaystyle {\ overline {S_ {1} S_ {2}}}}$ Main axis ${\ displaystyle {\ overline {S_ {3} S_ {4}}}}$ Minor axis a Major semi-axis b Small semi-axis F 1 , F 2 Focal point (ellipse) e lin. eccentricity M. Focus p Parameters (semi-latus rectum)

The  two characteristic radii of an ellipse are called semiaxes :

• The major semi-axis is half of the largest diameter of an ellipse, which is also called the main axis .
• The small semiaxis is half of the shortest diameter ( minor axis ) and is exactly at an angle of 90 ° to the major semiaxis.

The circle is a special ellipse in which these two semi-axes are the same length, in this case both semi-axes correspond to the radius of the circle.

The major axis (the largest diameter, here ) and the minor axis (the smallest diameter, here ) are collectively referred to as the major axes of the ellipse. Major and minor axes are conjugate diameters . This relationship is retained even when the ellipse is viewed "obliquely", which can be used for the geometric construction of other conjugate diameters. ${\ displaystyle {\ overline {S_ {1} S_ {2}}}}$${\ displaystyle {\ overline {S_ {3} S_ {4}}}}$

## astronomy

In astronomy , the semi-major axis of a Keplerian orbit is one of the six so-called orbital elements and is often given imprecisely as the “mean distance” and is usually abbreviated with a . It characterizes - together with the eccentricity  - the shape of elliptical orbits of different celestial bodies.

Such bodies are primarily the planets and their moons , artificial earth satellites , the asteroids and thousands of binary stars .

According to Kepler's third law , the orbital time U of an elliptical orbit is coupled with a ( ). The constant is related to the mass of the central body - in a planetary system, therefore, to the mass of the central star . ${\ displaystyle U ^ {2} / a ^ {3} = \ mathrm {const}}$

The two main vertices are called apses , the main axis is the apsidal line : If a body lies in the focal point F 1 and a smaller body circles it on an ellipse, the shortest distance ( = a - e ) is called the periapsis and the longest distance ( = a + e ) from the apoapsis ( perihelion, aphelion at the sun). ${\ displaystyle {\ overline {S_ {1} F_ {1}}}}$${\ displaystyle {\ overline {S_ {2} F_ {1}}}}$

In the periapsis (pericenter, main vertex close to the gravic center) the orbital velocity is maximal, in the apocenter minimal.

The actual mean distance depends not only on the major semi-axis but also on the numerical eccentricity and amounts to ${\ displaystyle \ varepsilon = e / a}$

${\ displaystyle a \ cdot \ left (1 + {\ frac {\ varepsilon ^ {2}} {2}} \ right)}$

## geodesy

In geodesy , the axes of the so-called error ellipses are an important means of representing the mean or maximum / minimum point errors. In the adjustment of geodetic networks , the leaves accuracy with which the individual measuring points of the network are determined, represent an error ellipse.

## Individual evidence

1. Erwin Groten: On the definition of the mean point error . In: Zeitschrift für Vermessungswesen (ZfV), 11/1969, pp. 455–457.