Hyperboloid

Parametric representations

Cartesian coordinates for the hyperboloids can be defined, similar to spherical coordinates, keeping the azimuth angle θ ∈ [0, 2π), but changing inclination v into hyperbolic trigonometric functions:

One-surface hyperboloid: v ∈ (−∞, ∞)

x = a cosh ⁡ v cos ⁡ θ y = b cosh ⁡ v sin ⁡ θ z = c sinh ⁡ v

Two-surface hyperboloid: v ∈ [0, ∞)

x = a sinh ⁡ v cos ⁡ θ y = b sinh ⁡ v sin ⁡ θ z = ± c cosh ⁡ v

Lines on the surface

A hyperboloid of one sheet contains two pencils of lines. It is a doubly ruled surface.

If the hyperboloid has the equation x 2 a 2 + y 2 b 2 − z 2 c 2 = 1 then the lines

g α ± : x → ( t ) = ( a cos ⁡ α b sin ⁡ α 0 ) + t ⋅ ( − a sin ⁡ α b cos ⁡ α ± c )   , t ∈ R ,   0 ≤ α ≤ 2 π  

are contained in the surface.

In case of a = b the hyperboloid is a surface of revolution and can be generated by rotating one of the two lines g 0 + or g 0 − , which are skew to the rotation axis (see picture). The more common generation of a hyperboloid of revolution is rotating a hyperbola around its semi-minor axis (see picture).

Remark: A hyperboloid of two sheets is projectively equivalent to a hyperbolic paraboloid.

Plane sections

For simplicity the plane sections of the unit hyperboloid with equation   H 1 : x 2 + y 2 − z 2 = 1 are considered. Because a hyperboloid in general position is an affine image of the unit hyperboloid, the result applies to the general case, too.

A plane with a slope less than 1 (1 is the slope of the lines on the hyperboloid) intersects H 1 in an ellipse,

A plane with a slope equal to 1 containing the origin intersects H 1 in a pair of parallel lines,

A plane with a slope equal 1 not containing the origin intersects H 1 in a parabola,

A tangential plane intersects H 1 in a pair of intersecting lines,

A non-tangential plane with a slope greater than 1 intersects H 1 in a hyperbola.

Properties of a hyperboloid of two sheets

The hyperboloid of two sheets does not contain lines. The discussion of plane sections can be performed for the unit hyperboloid of two sheets with equation

H 2 :   x 2 + y 2 − z 2 = − 1 .

which can be generated by a rotating hyperbola around one of its axes (the one that cuts the hyperbola)

A plane with slope less than 1 (1 is the slope of the asymptotes of the generating hyperbola) intersects H 2 either in an ellipse or in a point or not at all,

A plane with slope equal to 1 containing the origin (midpoint of the hyperboloid) does not intersect H 2 ,

A plane with slope equal to 1 not containing the origin intersects H 2 in a parabola,

A plane with slope greater than 1 intersects H 2 in a hyperbola.

Remark: A hyperboloid of two sheets is projectively equivalent to a sphere.

Common parametric representation

The following parametric representation includes hyperboloids of one sheet, two sheets, and their common boundary cone:

x → ( s , t ) = ( a s 2 + d cos ⁡ t b s 2 + d sin ⁡ t c s )

For d = 1 one gets a hyperboloid of one sheet,

for d = − 1 a hyperboloid of two sheets, and

for d = 0 a double cone.

Symmetries of a hyperboloid

The hyperboloids with equations x 2 a 2 + y 2 b 2 − z 2 c 2 = 1 , x 2 a 2 + y 2 b 2 − z 2 c 2 = − 1   are

pointsymmetric to the origin,

symmetric to the coordinate planes and

rotational symmetric to the z-axis and symmetric to any plane containing the z-axis, in case of a = b .

On the curvature of a hyperboloid

Whereas the Gaussian curvature of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive. In spite of its positive curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a model for hyperbolic geometry.

Generalised equations

More generally, an arbitrarily oriented hyperboloid, centered at v, is defined by the equation

( x − v ) T A ( x − v ) = 1 ,

where A is a matrix and x, v are vectors.

The eigenvectors of A define the principal directions of the hyperboloid and the eigenvalues of A are the reciprocals of the squares of the semi-axes: 1 / a 2 , 1 / b 2 and 1 / c 2 . The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue. The two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues.

In more than three dimensions

Imaginary hyperboloids are frequently found in mathematics of higher dimensions. For example, in a pseudo-Euclidean space one has the use of a quadratic form:

q ( x ) = ( x 1 2 + ⋯ + x k 2 ) − ( x k + 1 2 + ⋯ + x n 2 ) , k < n .

When c is any constant, then the part of the space given by

{ x   :   q ( x ) = c }

is called a hyperboloid. The degenerate case corresponds to c = 0.

As an example, consider the following passage:

... the velocity vectors always lie on a surface which Minkowski calls a four-dimensional hyperboloid since, expressed in terms of purely real coordinates (y₁, ..., y₄), its equation is y2
1 + y2
2 + y2
3 − y2
4 = −1, analogous to the hyperboloid y2
1 + y2
2 − y2
3 = −1 of three-dimensional space.

However, the term quasi-sphere is also used in this context since the sphere and hyperboloid have some commonality (See § Relation to the sphere below).

Hyperboloid structures

One-sheeted hyperboloids are used in construction, with the structures called hyperboloid structures. A hyperboloid is a doubly ruled surface; thus, it can be built with straight steel beams, producing a strong structure at a lower cost than other methods. Examples include cooling towers, especially of power stations, and many other structures.

Relation to the sphere

In 1853 William Rowan Hamilton published his Lectures on Quaternions which included presentation of biquaternions. The following passage from page 673 shows how Hamilton uses biquaternion algebra and vectors from quaternions to produce hyperboloids from the equation of a sphere:

In this passage S is the operator giving the scalar part of a quaternion, and T is the "tensor", now called norm, of a quaternion.

A modern view of the unification of the sphere and hyperboloid uses the idea of a conic section as a slice of a quadratic form. Instead of a conical surface, one requires conical hypersurfaces in four-dimensional space with points p = (w, x, y, z) ∈ R⁴ determined by quadratic forms. First consider the conical hypersurface

P = { p   :   w 2 = x 2 + y 2 + z 2 } and H r = { p   :   w = r } , which is a hyperplane.

Then P ∩ H r is the sphere with radius r. On the other hand, the conical hypersurface

Q = { p   :   w 2 + z 2 = x 2 + y 2 } provides that Q ∩ H r is a hyperboloid.

In the theory of quadratic forms, a unit quasi-sphere is the subset of a quadratic space X consisting of the x ∈ X such that the quadratic norm of x is one.