The Ellipsoid

The Definition of the Ellipsoid

The Standard Equation of the Ellipsoid

The Meaning of the Axes

• $a$: the semi-axis length of the ellipsoid in the $x$-direction
• $b$: the semi-axis length of the ellipsoid in the $y$-direction
• $c$: the semi-axis length of the ellipsoid in the $z$-direction

The Properties of the Ellipsoid

Symmetry

The ellipsoid is symmetric about the three coordinate planes:

• symmetric about the $yz$-plane (the plane $x = 0$)
• symmetric about the $xz$-plane (the plane $y = 0$)
• symmetric about the $xy$-plane (the plane $z = 0$)

The Vertices

The vertices of the ellipsoid on the coordinate axes are:

• $(a, 0, 0)$ and $(-a, 0, 0)$
• $(0, b, 0)$ and $(0, -b, 0)$
• $(0, 0, c)$ and $(0, 0, -c)$

The Nature of the Sections

The Volume of the Ellipsoid

The Surface Area

The surface area of an ellipsoid has no simple analytical expression, and it has to be computed numerically.

Special Cases

The Sphere

When $a = b = c = r$, the ellipsoid becomes a sphere:

$\frac{x^2}{r^2} + \frac{y^2}{r^2} + \frac{z^2}{r^2} = 1 \Rightarrow x^2 + y^2 + z^2 = r^2$

The Ellipsoid of Revolution

When two of the semi-axes are equal, an ellipsoid of revolution is formed:

• $a = b \neq c$: the ellipsoid formed by rotating about the $z$-axis
• $a = c \neq b$: the ellipsoid formed by rotating about the $y$-axis
• $b = c \neq a$: the ellipsoid formed by rotating about the $x$-axis

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Exercises

Exercise 1

Determine the type of the surface $\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1$ and find its semi-axis lengths.

Exercise 2

Find the volume of the ellipsoid $\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1$.

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Summary

Symbols Used in This Article