Ellipsoid
The earth is a spheroid shrink ( greatly exaggerated ).
The rugby ball has the shape of a prolate ellipsoid of revolution
An ellipsoid is the three - or multi-dimensional equivalent of an ellipse.
Known approximate examples of spheroids are the Earth, Jupiter and the rugby ball.
Definition
An ellipsoid in three-dimensional space can be explained as stretched or stunted ( affine ) image of a sphere surface ( sphere ). Using cartesian coordinates and orientation of the coordinate axes x, y and z is in accordance with the axes of symmetry of the ellipsoid 's equation
With positive real numbers, and the lengths of the semiaxes.
In the n- dimensional space is an ellipsoid
The amount of solution of a quadratic equation with real positive definite ( belonging to a square shape ) matrix.
Through main axis transform can be transformed to a diagonal matrix with positive eigenvalues. The eigenvectors of this matrix indicate the direction of the principal axes, the reciprocals of the roots of the eigenvalues are the lengths of the associated semi-axes.
In linear optimization ellipsoids are used in the ellipsoid method.
The following discussion is limited again on ellipsoids in three-dimensional space. If all three semiaxes different, it is called triaxial (or triaxial ) ellipsoid. Upon rotation of an ellipse about one of its axes a rotary body caused, in this case spheroids. Approximate examples of ellipsoids are rotating celestial bodies, such as the earth (cf. Erdellipsoid ) or other planets, suns or galaxies. Elliptical galaxies can also be triaxial.
Earth as an ellipsoid
The earth is only about a ball. In fact, it is flattened by rotation around itself at the poles and otherwise very irregularly shaped. To describe this in more detail irregularity, an ellipsoid of revolution is used instead of the ball frequently. This is used in cartography and geodesy as a reference system for the construction of measurement networks and direct indication of geographical coordinates. Through the figure of the earth ellipsoid is approximated as a surface of constant height (see geoid and sea level).
The planet Jupiter is due to its fast rotation (and because it is made mostly of gases and liquefied gases ) a clearly recognizable spheroid.
Volume of the ellipsoid
The volume can be set via
Calculate the product of the half-axles.
Surface of the ellipsoid of revolution
Is and is the eccentricity of the ellipse is calculated as the intersection with the XZ plane. Then, for a flattened ( oblate ) ellipsoid ( = axis of rotation Z -axis)
And an extended ( prolates ) ellipsoid ( = axis of rotation x-axis)
Surface of the triaxial ellipsoid
The surface of the triaxial ellipsoid can not be expressed by means of functions that one sees as fundamental, such as artanh or arcsin. The area calculation Adrien -Marie Legendre succeeded with the help of the elliptic integrals. Be. If we write
These are the integrals
The surface has E and F according to the value of Legendre
If the expressions for k and as well as the substitutions
Used in the equation for A, so there is the notation
From Knud Thomsen comes the (integral -free ) approximation formula
The maximum deviation from the exact result is less than 1.2 %.
In the limiting case of a completely flattened ellipsoid all three formulas given for A strive to twice the area of an ellipse with semi-axes and.
Derivation of the formulas for ellipsoids of revolution
With the definitions of the elliptic integrals E and F, the two rotationally symmetric special cases can be easily derived from the general triaxial formula, because E and F are to elementary functions.
Oblate ellipsoid
Extended ellipsoid
Summarizing and simplifying leads to the indicated section of the ellipsoid surface expressions. Alternatively, the surfaces as lateral surfaces of rotating ellipses ( spheroid ) can be calculated.