Ellipsoid

An ellipsoid is a special shape made by changing a sphere. You can stretch or squeeze the sphere in different directions to make it. This can also be done using a process called an affine transformation.

Ellipsoids are part of a group called quadric surfaces. This means they can be described using a special kind of math equation. One important feature of an ellipsoid is that if you cut through it with a flat surface, the shape you see on the cut is always an ellipse, or sometimes just a single point. An ellipsoid is a closed shape that can fit inside a big enough sphere.

Ellipsoids have three special straight lines called axes. These axes cross at a central point and are all at right angles to each other. The parts of these axes on the surface of the ellipsoid are called the main axes. When the three axes are all different lengths, the shape is called a triaxial ellipsoid. If two axes are the same length, the ellipsoid is called an ellipsoid of revolution, or a spheroid. This means it looks the same after you spin it around one of its axes. If the third axis is shorter, it is called an oblate spheroid, which is like a squashed sphere. If the third axis is longer, it is called a prolate spheroid, which is like a stretched sphere. If all three axes are the same length, the ellipsoid is simply a sphere.

Standard equation

An ellipsoid is a special shape that looks like a sphere that has been gently stretched or squished in different directions. We can describe this shape using a simple math rule.

If we imagine a point in space with coordinates (x, y, z), the point is on the surface of an ellipsoid when this equation is true:

^x² / ^a² + ^y² / ^b² + ^z² / ^c² = 1

Here, a, b, and c are numbers that tell us how long the shape is along each direction. When all three numbers are the same, the shape is a perfect sphere. If two numbers are the same and the third is different, we get a special type of ellipsoid called a spheroid.

Volume

The volume of an ellipsoid can be found with a simple formula. If we know the lengths of three important measurements, called radii (a, b, and c), the volume V is:

V = ⁴/ ₃ π a b c

If all three radii are the same, the shape becomes a sphere, and the formula matches the volume of a sphere. When two radii are equal, the shape is called an oblate or prolate spheroid.

The volume of an ellipsoid also relates to the volumes of certain boxes that can fit around it. The volume of the smallest box that fits around the ellipsoid is eight times the volume of the ellipsoid, while the volume of the largest box that can fit inside it is a bit smaller.

Surface area

Plane sections

Determining the ellipse of a plane section

To find the shape of the ellipse made when a flat surface cuts through an ellipsoid, we start with the equation of the ellipsoid and the equation of the flat surface. We then find three special points that help us draw the ellipse.

The article also includes an example showing how this works with specific numbers.

How to find the points and lengths that define the ellipse is described in ellipse.

Pins-and-string construction

The pins-and-string method builds an ellipsoid shape using ideas from making an ellipse with pins and string.

For an ellipsoid made by rotating an ellipse, the same pins-and-string idea works. Making a more complex shape called a triaxial ellipsoid is trickier. Early ideas came from a Scottish scientist named J. C. Maxwell in 1868. German mathematician O. Staude did more work on this in later years. A book called Geometry and the Imagination describes this method.

Steps of the construction

Choose an ellipse E and a hyperbola H, which are a pair of focal conics.
Pin one end of the string to a point on the ellipse and the other to a focus. Keep the string tight at a point P.
P is a point of the ellipsoid.
The remaining points of the ellipsoid can be found by moving the string.

Semi-axes

Equations can be used to find the sizes of the ellipsoid.

Converse

If an ellipsoid is given, we can find the details for a pins-and-string construction.

Confocal ellipsoids

If an ellipsoid is confocal to another, they share the same focal conics.

Limit case, ellipsoid of revolution

In a special case, the ellipsoid is symmetric around an axis.

Properties of the focal hyperbola

If you view an ellipsoid from a certain point, it can look like a sphere.

Umbilical points

The focal hyperbola meets the ellipsoid at special points.

Property of the focal ellipse

The focal ellipse can be thought of as a very thin ellipsoid.

In higher dimensions and general position

A hyperellipsoid is a special shape that exists in more than three dimensions. You can imagine it as a stretched or squished sphere. This stretching happens through a process called an affine transformation, which includes moving, turning, and changing the size of the shape.

We can use math to describe the size and shape of a hyperellipsoid. One way to think about it is that it starts as a regular sphere and then gets stretched in different directions. Math also helps us find the space these shapes take up in different dimensions.

Applications

Ellipsoids are useful in many areas:

Geodesy

Earth ellipsoid helps us understand the shape of the Earth.
Reference ellipsoid is also used to study other planets.

Mechanics

Poinsot's ellipsoid helps us see how spinning objects move.
Lamé's stress ellipsoid shows how forces act on materials.
Manipulability ellipsoid describes how robots move.
Jacobi ellipsoid is a special shape formed by spinning liquid.

Crystallography

Index ellipsoid shows how light bends in crystals.
Thermal ellipsoid shows how atoms move in crystals.

Artist's conception of Haumea, a Jacobi-ellipsoid dwarf planet, with its two moons

Computer science

Ellipsoid method is an important math tool.

Lighting

Medicine

MRI can measure the size of certain body parts using an ellipsoid shape.

Dynamical properties Ellipsoids spin steadily around their longest or shortest axis. This is why some space objects, like Haumea, spin this way.

Fluid dynamics Ellipsoids are used to study how fluids flow around objects. This helps us understand things like tiny particles moving in water.

Probability and statistics Ellipsoids help describe patterns in data, especially in finance and other fields.