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The semi-major axis (major semiaxis) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis (minor semiaxis) 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.
A reference ellipsoid, customarily chosen to be the same size (volume) as the geoid, is described by its semi-major axis (equatorial radius) a and flattening f. The quantity f = (a−b)/a, where b is the semi-minor axis (polar radius), is a purely geometrical one.
where and are the lengths of the semi-major and semi-minor axes, respectively. The area formula π a b {\displaystyle \pi ab} is intuitive: start with a circle of radius b {\displaystyle b} (so its area is π b 2 {\displaystyle \pi b^{2}} ) and stretch it by a factor a / b {\displaystyle a/b} to make an ellipse.
The parameters determined are usually the semi-major axis, , and any of the semi-minor axis, , flattening, or eccentricity. Regional-scale systematic effects observed in the radius of curvature measurements reflect the geoid undulation and the deflection of the vertical, as explored in astrogeodetic leveling.
They correspond to the semi-major axis and semi-minor axis of an ellipse. In spherical coordinate system for which ( x , y , z ) = ( r sin θ cos φ , r sin θ sin φ , r cos θ ) {\displaystyle (x,y,z)=(r\sin \theta \cos \varphi ,r\sin \theta \sin \varphi ,r\cos \theta )} , the general ellipsoid is defined as:
When increases from zero, i.e., assumes positive values, the line evolves into an ellipse that is being traced out in the counterclockwise direction (looking in the direction of the propagating wave); this then corresponds to left-handed elliptical polarization; the semi-major axis is now oriented at an angle .
The semi major axis is not immediately visible with a hyperbolic trajectory but can be constructed as it is the distance from periapsis to the point where the two asymptotes cross. Usually, by convention, it is negative, to keep various equations consistent with elliptical orbits.
where a is the semi-major axis and b is the semi-minor axis. For a point on the ellipse, P = P(x, y), representing the position of an orbiting body in an elliptical orbit, the eccentric anomaly is the angle E in the figure.