# What does ellipsoid

**What is an ellipsoid?**

... | An ellipsoid is the graph of the relation x² / a² + y² / b² + z² / c² = 1. The largest possible domain is D = {(x, y, z) | -a <= x <= a, -b <= y <= b, -c <= z <= c} It is shown here in the Cartesian coordinate system. For the drawing, x² / 4 + y² / 2 + z² / 1 = 1 applies. |

... | The parametric representation has the same ellipsoid x = a * sin (u) cos (t) y = b * sin (u) sin (t) z = c * cos (u) D = {(t, u) | -2pi; <= t <= 2pi, -pi <= u <= pi} |

It really is the same ellipsoid as the following calculation shows.

The following applies: x² / a² + y² / b² + z² / c²

= [a * sin (u) cos (t)] ² / a² + [b * sin (u) sin (t)] ² / b² + [c * cos (u)] ² / c² = sin² (u) cos² (t) + sin² (u) sin² (t) + cos² (u) = sin² (u) + cos² (u) = 1.

On my page Torus I explain how to draw bodies of this type with the program Winplot.

... | The ellipsoid is the generalization of the ellipse to the third dimension. This is the graph of the relation x² / a² + y² / b² = 1 with D = {(x, y) | -a <= x <= a, -b <= y <= b}. |

**Designations **Top

If you put a coordinate equal to zero in the coordinate equation x² / a² + y² / b² + z² / c² = 1, you get the three *Main ellipses* of the ellipsoid.

x = 0 leads to the ellipse equation y² / b² + z² / c² = 1 with regard to the y-z plane | y = 0 leads to the ellipse equation x² / a² + z² / c² = 1 with regard to the x-z plane | z = 0 leads to the ellipse equation x² / a² + y² / b² = 1 with regard to the x-y plane |

By the way, every plane section through the ellipsoid leads to an ellipse.

The ellipsoid is symmetrical to the main planes, since the term of the relation does not change if one replaces x, y and z with their opposite numbers -x, -y and -z.

... | The radiuses a, b and c of the main ellipses are also called Radius of the ellipsoid.If a> b> c, then a is the The vertices of the main ellipses are also called |

An ellipsoid with a> b> c is called a three-axis ellipsoid.

**Ellipsoids of revolution **Top

Ellipsoids, where two out of three radiuses are the same, play a special role. For example, let a = b.

Then the equation (x² + y²) / a² + z² / c² = 1.

A distinction is made between two cases, namely c> a or c

...... | The body on the left has the equation (x² + y²) / a² + z² / c² = 1, where c> a is.If one sets z = 0, the equation becomes x² + y² = a². Rugby balls, for example, have this shape. |

Flattened ellipsoid or spheroid

So one differentiates between the following ellipsoids.

a | a = b, a | a = b, a> c | a = b = c |

**Volume and surface **Top

The volume and the surface of ellipsoids of revolution can be calculated elementarily.

If a curve rotates around the x-axis with y = f (x), the following two formulas apply for volume and surface area.

Elongated ellipsoid

volume

...... | So imagine that the ellipse with x² / a² + y² / b² = 1 or y² = (b² / a²) (a²-x²) rotates around the x-axis. Then applies |

surface

To calculate the second integral, one sets

> the equation x² / a² + y² / b² = 1,

> the derivative 2x / a² + 2 (y / b²) y '= 0 or y' = - (2b²x) / (2a²y) or y'² = (b

^{4}x

^{2}) / (a

^{4}y

^{2}) and

> the term sqrt (1 + y'²) = sqrt [1+ (b

^{4}x

^{2}) / (a

^{4}y

^{2})] = ... = (b / y) sqrt [1- (ex / a²) ²] with e² = a²-b² ready.

That leads to

The substitution is called z = ex / a², and it is dz = (e / a²) dx or dx = (a² / e) dz.

According to Bronstein (4, page 47) there is the following antiderivative for the integral.

Then

Result

V = (4/3) pi * ab² and O = 2pi * b [b + a² / e * arc sin (e / a)] with e² = a²-b²

Flattened ellipsoid or spheroid

Triaxial ellipsoid

Ellipsoids are generally given by the semi-axes a, b and c. From this, the volume and the surface can be calculated. - I'll leave it at the formulas, for the surface I have to limit myself to an approximation.

found at de.wikipedia, more is available from Gérard P. Michon (URL below) |

**Second order surfaces **Top

The ellipsoid belongs to the second order surfaces.

They result when the equation Ax + By + Cz + Dxy + Eyz + Fzx + Gx + Hy + Kz + L = 0 is graphically represented in a spatial Cartesian coordinate system.

With certain values for the variables A to L, the following areas essentially result.

**Ellipsoid on the internet **Top

German

Lok Lam Mak (technical work by a 12th grade student)

The volume of a chicken egg

PiMath

THE SHAPE OF THE EARTH

Ralf Schaper (Department of Mathematics / Computer Science, University of Kassel)

Cut ellipsoid

Wikipedia

Ellipsoid, ellipsoid of revolution

English

F. W. PRESTON

The Volume of an Egg

Eric W. Weisstein (MathWorld)

Ellipsoid, Spheroid, Quadratic Surface, Superegg

Gerard P. Michon

Spheroids & Scalene Ellipsoids

Marlene Dieguez Fernandez

Quadric Surfaces

Richard Parris (freeware program WINPLOT)

The official website is closed. Download the German program e.g. from heise

The Wolfram Demonstrations Project

Ellipsoid, Eggnigmatica, EggnigmaticaII, Lamé's Ellipsoid And Mohr's Circles

Wikipedia

Ellipsoid, spheroid

Xahlee

Ellipsoid, rotate me

French

Robert FERRÉOL (mathcurve)

Ellipsoids

**credentials **Top

(1) Georg Ulrich / Paul Hoffmann: Differential and integral calculus for self-teaching, Hollfeld [ISBN 3 8044 0575 4]

Message from Torsten Sillke:

(2) Carlson, B.C., Special Functions of Applied Mathematics, pp. 261-272, Academic Press, New York (NY), 1977. N.B. In example 9.4-2, Carlson presents an elegant derivation of the surface area of an ellipsoid in standard function form.

**Feedback:**Email address on my main page

URL of my homepage:

http://www.mathematische-basteleien.de/

© 2009 Jürgen Köller

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