Ellipsoid: features and examples

The ellipsoid is a surface in space that belongs to the quadratic surface group and whose general equation is as follows:

Ax 2 + By 2 + Cz 2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0

It is the three-dimensional equivalent of an ellipse, characterized by having elliptical and circular features in some special cases. Traces are the curves obtained by the intersection of the ellipsoid with a plane.

In addition to the ellipsoid, there are five more quadratics: one- and two-leaf hyperboloid, two types of paraboloid (hyperbolic and elliptical), and the elliptical cone. Its features are also conical.

The ellipsoid can also be expressed using the standard equation in Cartesian coordinates. An ellipsoid centered at the origin (0,0,0) and expressed in this way resembles an ellipse, but with an additional term:

Values a , b, and c are real numbers greater than 0 and represent the three half axes of the ellipsoid.

Ellipsoid Characteristics

– Standard equation 

The standard equation in Cartesian coordinates for the ellipse centered on the point (h, k, m) is:

– Parametric ellipsoid equations

In spherical coordinates, the ellipsoid can be described as follows:

x = a sin θ. why

y = b sin θ. if you are

z = c cos θ

The semi-axes of the ellipsoid are still a, b and c, while the parameters are the angles θ and φ in the following figure:

– Ellipsoid traces

The general equation for a surface in space is F (x, y, z) = 0 and the surface traces are the curves:

– x = c; F(c,y,z) = 0

– y = c; F(x,c,z) = 0

– z = c; F(x,y,c) = 0

In the case of an ellipsoid, these curves are ellipses and sometimes circles.

– Volume

The volume V of the ellipsoid is given by (4/3) π times the product of its three semi-axes:

V = (4/3) π. ABC

Ellipsoid Special Cases

-An ellipsoid becomes a sphere when all semi-axes are the same size: a = b = c ≠ 0. This makes sense, since the ellipsoid is like a sphere that has been stretched differently along each axis.

-The spheroid is an ellipsoid in which two of the semi-axes are identical and the third is different, for example it could be a = b ≠ c.

The spheroid is also called the ellipsoid of revolution because it can be generated by rotating ellipses around an axis.

If the axis of rotation coincides with the main axis, the spheroid is prolate , but if it coincides with the minor axis, it is oblate :

The measurement of the flatness of the spheroid (ellipticity) is given by the difference in length between the two half-axes, expressed in a fractional way, that is, it is the flatness of the unit, given by:

f = (a – b) / a

In this equation, a represents the semi-major axis and b the semi-minor axis, remember that the third axis is equal to one of these for a spheroid. The value of f is between 0 and 1, and for a spheroid it must be greater than 0 (if it were equal to 0, we would simply have a sphere).

The reference ellipsoid

Planets and generally stars are not usually perfect spheres, because rotational movement around their axes flattens the body at the poles and increases the volume at the equator.

This is why the Earth looks like an oblate spheroid, although it is not as exaggerated as the one in the previous figure, and in turn, the gas giant Saturn is the flattest of the planets in the solar system.

Therefore, a more realistic way to represent the planets is to assume that they are like a spheroid or ellipsoid of revolution, whose semi-major axis is the equatorial radius and the semi-minor axis is the polar radius.

Careful measurements made on the globe made it possible to construct the Earth’s reference ellipsoid as its most accurate way of working mathematically.

Stars also have rotational motions that give them more or less flattened shapes. The fast star Achernar, the eighth brightest star in the night sky, in the southern constellation, Eridanus is remarkably elliptical compared to most. That’s 144 light years from us.

At the other extreme, scientists a few years ago found the most spherical object found so far: the star Kepler 11145123, 5000 light-years away, twice the size of our Sun and a half-axis difference of just 3 km . As expected, it also rotates slower.

As for Earth, it’s also not a perfect spheroid because of its rugged surface and local variations in gravity. Therefore, there is more than one reference spheroid available and the one most suitable for the local geography is chosen at each location.

The help of satellites is invaluable in creating ever more accurate models of the Earth’s shape, thanks to them, for example, it is known that the south pole is closer to the equator than the north pole.

Numerical example

Due to the Earth’s rotation, a centrifugal force is generated which gives it the shape of an oblong ellipsoid instead of a sphere. The Earth’s equatorial radius is known to be 3963 miles and the polar radius is 3942 miles.

Find the equation of the equatorial trace, that of the ellipsoid and the measure of its flatness. Also compare with the ellipticity of Saturn, with the data given below:

-Equatorial radius of Saturn: 60 268 km

Saturn’s polar radius: 54,364 km

Related:   Orion Nebula: origin, location, characteristics and data

Solution

A coordinate system is needed, which we will assume centered on the origin (earth center). We will assume that the vertical z axis and the trace corresponding to the equator are in the xy plane, equivalent to the z = 0 plane.

In the equatorial plane, the semi-axes a and b are equal, so a = b = 3963 miles, while c = 3942 miles. This is a special case: a spheroid centered on the point (0,0,0) as indicated above.

The equatorial trace is a circumference of radius R = 3963 miles, centered on the origin. It is calculated by setting z = 0 in the standard equation:

And the standard equation for the terrestrial ellipsoid is:

f)  Land = (a – b) / a = (3963-3942) miles / 3963 miles = 0.0053

Saturn = (60268-54363) km / 60268 km = 0.0980

Note that the ellipticity f is a dimensionless quantity.

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