Spheroid

An ellipsoid of revolution (English spheroid ) is an ellipsoid, which is formed by rotation of an ellipse about one of its axes. In contrast to a general ellipsoid two axles have the same length. A distinction is made depending on the length of the axis of rotation

• Extended ( prolate ) ellipsoid in rotation about the major axis A and the
• Flattened ( oblate ) ellipsoid in rotation around the semi-minor axis b.

An example of an elongated ellipsoid of revolution, the shape of rugby ball, which is similar to a flattened chocolate lens.

Occurrence

Most major celestial bodies are oblate spheroids approximated (also called spheroids ). They are created by the centrifugal force, which causes a rotating spherical body is deformed. At the poles, so the intersection points of the axis of rotation, these bodies are flattened at the equator creates a bulge. Particularly pronounced flattening at the large gas planets Jupiter and Saturn is pronounced, because they rotate very quickly and are not solidified. But the Earth and the other terrestrial planets are deformed by the centrifugal force resulting from the rotation to rotation ellipsoid. The rotating in ten hours Jupiter is flattened to about 1 /16, the Earth flattening is 1/ 298 257 223 563 ( WGS 84). Elliptical galaxies are often no spheroids, but triaxial.

Parameter representation

The constant stretching factors a, b and c in this case describe the respective point of intersection with the associated axis of the Cartesian coordinate system. On the parametric representation of the relation to the ball clear. A = b = c, we obtain spherical coordinates, ie the ball is a special case of an ellipsoid of revolution.

Volume

The volume is of the elongated ellipsoid of revolution

And that of the flattened

Half- axis to the line through the two poles of the rotating body respectively appear as a linear factor.

Surface

The surface of the elongated ellipsoid, we can calculate with

That of the oblate with

Application

In geodesy, cartography and other geosciences spheroids are used as geometric approach to the (physical ) geoid. These spheroids are then used as a reference surface to specify the position or height of objects of the earth's surface. This is called a reference ellipsoid.

In a hollow body, the boundary surfaces of the ( elongated ) spheroids reflect the radiation from a focal point to the other. The effect uses a Flüstergewölbe for the bundling of sound waves. Thus formed optical reflectors focus the radiation from a nearly point-like, is located in a focal point of the light source to the other focal point of the ellipsoid. There, the boundary surface of a light pipe, to another optical element, or the location of a radiation-induced process may be located.

In the sport of rugby ball has the shape of an elongated ellipsoid of revolution, the (classical ) discussion takes the form of an oblate spheroid.

Formulas to calculate the surface

Be the equation of the ellipse with semi-axes a and b ( a> b).

Extended spheroid

With the first guldinschen control can calculate the surface of the rotating body, which is generated by rotating about the x- axis of the ellipse. This one takes as a generating line, resulting from the ellipse equation by solving for y.

In addition, an integral is still needed, which can be verified via the derivative of the right-hand side with respect to x:

Provides use of F and F ' in the control guldinsche

In the last expression, the mirror symmetry of the integrand has been exploited by x = 0. Now all you need is the pq -integral with the upper limit ( the lower limit does not contribute ) and using the parameters and and gets

Which after simplification is the desired formula.

Oblate spheroid

This calculation is similar to the previous example but now allowing the ellipse rotate around the y-axis. is the interval of the considered line of profile ( here [0, a] ). With the first guldinschen usually in the form

And the inverse function ( dissolving the ellipse equation for ), and use of, and one finds

Here again, the symmetry of the ellipse is used, and also the integration limits disposed such that a positive result.

Substituting in the pq -integral, we obtain the right at this point related form

Insertion of the integral with the upper limit and the parameters and results in

Simplify the above-cited expression provides for the surface of the oblate spheroid. The result can be calculated using the numerical eccentricity

Pose and with the replacement

Bring in a form commonly used.

Both surfaces of formulas are for interchanging of a and b (corresponding to a rotation of the spheroid forming the ellipse to 90 degrees) into each other.