Ellipsoid

An ellipsoid is a special shape that comes from changing a sphere by stretching or squeezing it in different directions. This can also be done using a process called an affine transformation.

Ellipsoids are part of a group called quadric surfaces, which 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. Also, an ellipsoid is a closed shape that can fit inside a big enough sphere.

Ellipsoids have three special straight lines called axes that cross at a central point. These axes are all at right angles to each other. The parts of these axes that lie 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. And if all three axes are the same length, the ellipsoid is simply a sphere.

Standard equation

An ellipsoid is a special kind of 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 calculated using 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

This means that 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 a specific 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 process involves changing the coordinates to make the problem simpler, finding the center and size of the circle on a sphere, and then changing back to the original coordinates to get the ellipse on the ellipsoid.

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 by using ideas from making an ellipse with pins and string (see diagram).

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 1882, 1886, and 1898. A book called Geometry and the Imagination by Hilbert and Cohn-Vossen describes this method.

Steps of the construction

Choose an ellipse E and a hyperbola H, which are a pair of focal conics: E ( φ ) = ( a cos φ , b sin φ , 0 ) H ( ψ ) = ( c cosh ψ , 0 , b sinh ψ ), c² = a² − b² with the vertices and foci of the ellipse S₁ = ( a , 0 , 0 ), F₁ = ( c , 0 , 0 ), F₂ = ( − c , 0 , 0 ), S₂ = ( − a , 0 , 0 ) and a string (in diagram red) of length l.
Pin one end of the string to vertex S₁ and the other to focus F₂. The string is kept tight at a point P with positive y- and z-coordinates, such that the string runs from S₁ to P behind the upper part of the hyperbola (see diagram) and is free to slide on the hyperbola. The part of the string from P to F₂ runs and slides in front of the ellipse. The string runs through that point of the hyperbola, for which the distance |S₁P| over any hyperbola point is at a minimum. The analogous statement on the second part of the string and the ellipse has to be true, too.
Then: P is a point of the ellipsoid with equation x²/r_x² + y²/r_y² + z²/r_z² = 1 r_x = ½ ( l − a + c ), r_y = r_x² − c² , r_z = r_x² − a² .
The remaining points of the ellipsoid can be constructed by suitable changes of the string at the focal conics.

Semi-axes

Equations for the semi-axes of the generated ellipsoid can be derived by special choices for point P:

Y = ( 0 , r_y , 0 ), Z = ( 0 , 0 , r_z ).

The lower part of the diagram shows that F₁ and F₂ are the foci of the ellipse in the xy-plane, too. Hence, it is confocal to the given ellipse and the length of the string is l = 2r_x + (a − c). Solving for r_x yields r_x = ½(l − a + c); furthermore r_y² = r_x² − c².

From the upper diagram we see that S₁ and S₂ are the foci of the ellipse section of the ellipsoid in the xz-plane and that r_z² = r_x² − a².

Converse

If, conversely, a triaxial ellipsoid is given by its equation, then from the equations in step 3 one can derive the parameters a, b, l for a pins-and-string construction.

Confocal ellipsoids

If E is an ellipsoid confocal to E with the squares of its semi-axes

r̄_x² = r_x² − λ , r̄_y² = r_y² − λ , r̄_z² = r_z² − λ

then from the equations of E

r_x² − r_y² = c² , r_x² − r_z² = a² , r_y² − r_z² = a² − c² = b²

one finds, that the corresponding focal conics used for the pins-and-string construction have the same semi-axes a, b, c as ellipsoid E. Therefore (analogously to the foci of an ellipse) one considers the focal conics of a triaxial ellipsoid as the (infinite many) foci and calls them the focal curves of the ellipsoid.

The converse statement is true, too: if one chooses a second string of length l and defines

λ = r_x² − r̄_x²

then the equations

r̄_y² = r_y² − λ , r̄_z² = r_z² − λ

are valid, which means the two ellipsoids are confocal.

Limit case, ellipsoid of revolution

In case of a = c (a spheroid) one gets S₁ = F₁ and S₂ = F₂, which means that the focal ellipse degenerates to a line segment and the focal hyperbola collapses to two infinite line segments on the x-axis. The ellipsoid is rotationally symmetric around the x-axis and

r_x = ½l , r_y = r_z = r_x² − c² .

Properties of the focal hyperbola

True curve

If one views an ellipsoid from an external point V of its focal hyperbola, then it seems to be a sphere, that is its apparent shape is a circle. Equivalently, the tangents of the ellipsoid containing point V are the lines of a circular cone, whose axis of rotation is the tangent line of the hyperbola at V. If one allows the center V to disappear into infinity, one gets an orthogonal parallel projection with the corresponding asymptote of the focal hyperbola as its direction. The true curve of shape (tangent points) on the ellipsoid is not a circle.

The lower part of the diagram shows on the left a parallel projection of an ellipsoid (with semi-axes 60, 40, 30) along an asymptote and on the right a central projection with center V and main point H on the tangent of the hyperbola at point V. (H is the foot of the perpendicular from V onto the image plane.) For both projections the apparent shape is a circle. In the parallel case the image of the origin O is the circle's center; in the central case main point H is the center.

Umbilical points

The focal hyperbola intersects the ellipsoid at its four umbilical points.

Property of the focal ellipse

The focal ellipse together with its inner part can be considered as the limit surface (an infinitely thin ellipsoid) of the pencil of confocal ellipsoids determined by a, b for r_z → 0. For the limit case one gets

r_x = a , r_y = b , l = 3a − c .

In higher dimensions and general position

A hyperellipsoid is a special shape in higher dimensions. It can be thought of as a stretched or squished sphere. This stretching happens through a process called an affine transformation, which includes shifting, rotating, and scaling.

The size and shape of a hyperellipsoid can be described using special math formulas. One way to think about it is as the result of stretching a regular sphere in different directions. The math also lets us calculate the volume of these shapes in different dimensions.

Applications

Ellipsoids are useful in many areas:

Geodesy

Earth ellipsoid, a shape that helps us understand the Earth's form.
Reference ellipsoid, used to study other planets too.

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 have special ways of spinning. They 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, which helps in understanding things like tiny particles moving in water.

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